growth of a free group automorphism is same for finite index subgroups?

While I cannot point to any place in the literature with this particular statement, I would say that it is an exercise in relative train track theory, which is the very nice normal form theory for $\text{Out}(F_n)$ and $\text{Aut}(F_n)$ found in papers of Bestvina, Feighn, and Handel (in various subgroupings). Usually relative train track theory is applied to $\text{Out}(F_n)$ but you can apply it to $\text{Aut}(F_n)$ with a little trick.

First, you can embed $\text{Aut}(F_n) \hookrightarrow \text{Out}(F_{n+1})$ by letting $F_{n+1}=\langle a_0,a_1,...,a_n\rangle$, and restricting to the subgroup in which $a_0$ is fixed and $a_1,...,a_n$ are mapped to themselves. Every $\phi \in \text{Aut}(F_n)$ is represented by a relative train track map $f : G \to G$ where $G$ has a closed edge labelled $a_0$ with base point $p$ such that $a_0$ is fixed by $f$, and $f$ preserves $G \setminus H$, inducing the automorphism $\phi$ on $\pi_1(G \setminus H,p) \approx F_n=\langle a_1,...,a_n\rangle$. Let's refer to this map $f$ as a "relative train track representative of $\phi \in \text{Aut}(F_n)$" (this is a nonstandard terminology: ordinarily relative train maps are things which represent outer automorphisms, and this $\phi$ does indeed represent an outer automorphism of $F_{n+1}$, but we have used the "fixed loop trick" to make $\phi$ also represent an actual automorphism of $F_n$).

I still have to say what the "normal form" properties of $f$ are, i.e. I have to tell you a little bit about the definition of a relative train track map. After passing to a power, $G$ is written as a nested union $$G_0 \subset G_1 \subset ... \subset G_K=G $$ of $f$-invariant subgraphs, starting with $G_0=a_0$. For each subgraph $H_i = G_i - G_{i-1}$ there are three possibilities: $H_i$ is a "zero stratum" which means that $H_i$ is contractible and its image under $f$ is contained in $G_{i-1}$; or $H_i$ is an "NEG stratum" (meaning "non-exponentially growing") meaning that $H_i$ is a single edge $E$ such that $f(E)=Eu$ with $u$ some path in $G_{i-1}$; or $H_i$ is an "EG stratum" meaning that has $m \ge 2$ edges the $m \times m$ transition matrix of $f$ acting on the edges of $H_i$ is a Perron-Frobenius matrix, and some additional "non cancellation" properties hold which I won't bother to make explicit.

So, the major points are:

1. One can detect the growth type of $f$ from this normal form. For example, $\phi$ has exponential growth if and only if some stratum $H_i$ is an EG stratum. The NEG strata are so named because each crosses itself only once, but the NEG strata can generate polynomial growth by the manner in which they cross over lower stratum edges (think of an example on $F_3 = \langle a_1,a_2,a_3 \rangle$ where $a_1 \mapsto a_1$, $a_2 \mapsto a_2 a_1$, and $a_3 \mapsto a_3 a_2$; in this example, $a_3$ has quadratic growth). One outcome of this is that every automorphism of $F_n$ has either exponential growth or polynomial growth of some integer degree (between $0$ and $n-1$).
2. One can obtain a relative train track map representing $\phi | H \in \text{Aut}(H)$ from a covering space trick as follows. Construct a pointed covering space of $(G \setminus a_0,p)$ representing the subgroup $H$. Then attach a closed edge to each lift of $p_0$, obtaining a covering space $G'$ of $G$, a subgraph $A \subset G'$ which is the union of the lifts of $a_0$, and a base point $p' \in G'$ to which one of the loops $a'_0$ is attached, so that $(G' \setminus A,p')$ is the covering space of $(G \setminus a_0,p)$ representing $H$. The map $f$ lifts to a map $f' : G' \to G'$ fixing $p'$, and preserving $G' \setminus A$, so that the restiction of $f'$ to $(G' \setminus A,p')$ induces the homomorphism $\phi | H$. Thus, we may think of $f' : G' \to G'$ as a train track representative of $\phi | H$ (again a nonstandard terminology). One needs to check that $f'$ satisfies the relative train track axioms, using that $f$ satisfies those axioms, but this is straightforward. We can therefore use $f'$ to detect the growth properties of $\phi | H$.
3. If $\phi$ has exponential growth, then $G$ has an EG stratum $H_i=G_i \setminus G_{i-1}$, implying that the inverse image of $H_i$ in $G'$ is a (union of) EG strata of $f'$, implying that $\phi | H$ has exponential growth.
4. If $\phi$ has polynomial growth, equivalently $\phi$ does not have exponential growth, then every stratum of $G$ is an NEG stratum, implying that every stratum of $G'$ is an NEG stratum, implying that $\phi | H$ does not have exponential growth, implying that $\phi | H$ has polynomial growth.
5. I believe that one can argue similarly regarding degrees of polynomial growth, demonstrating that if $\phi$ has polynomial growth of degree $d$ then $\phi'$ has polynomial growth of degree $d$.

I have an idea that I believe worked out in detail would give a proof, although I haven't done this:

Without loss of generality we can assume that $H$ is a normal subgroup, because every finite-index subgroup contains a finite-index normal subgroup.

Then if $x$ and $y$ are two elements in the same class modulo $H$, then both $xy^{-1}$ and $y^{-1} x$ are contained in $H$, so we may bound $||\phi^n(xy^{-1})||$ and $||\phi^n(y^{-1}x)||$ using the growth function of $\phi$ restricted to $H$. This is useful, because if we write $x$ and $y$ as a string of symbols, all but $||\phi^n(xy^{-1})||$ symbols on the right side of $\phi^n(x)$ agree with all but $||\phi^n(xy^{-1})||$ symbols on the right side of $\phi^n(y)$, and similarly all but $||\phi^n(y^{-1}x)||$ symbols on the left side of $\phi^n(x)$ agree with all but $||\phi^n(y^{-1}x)||$ symbols on the left side of $\phi^n(y)$.

If the length of $\phi^n(x)$ is much larger than $||\phi^n(xy^{-1})||+||\phi^n(y^{-1}x)||$ then this implies that $\phi^n(x)$ and $\phi^n(y)$ have a peculiar structure. They each have a beginning string, an ending string, and between them a middle string repeated different numbers of times. The periodicity of the middle string comes from applying the equality of the left sides to get between $\phi^n(x)$ and $\phi^n(y)$ and then applying the equality of the right sides to get back, which represents a translation of $||\phi^n(x)|| - ||\phi^n(y) || \leq ||\phi^n(xy^{-1})||$ and preserves every symbol.

So if $\phi^n(x)$ is indeed much larger than the growth function of $\phi$ restricted to $H$, this should imply that for all $y$ sufficiently close to $x$ in the same class modulo $H$, $\phi^n(y) = g_1 g_2^k g_3$ for fixed $g_1,g_2,g_3$ and some $k$ depending on $y$. Hence we have $y = \phi^{-n} ( g_1) \phi^{-n}(g_2)^k \phi^{-n}(g_3)$ for all such $y$, which surely can't ahppen in the free group as long as we take the neighborhood large enough.

An alternate way to address point 5 in Lee Mosher's above answer, using only that $\phi$ and $\phi|_H$ have quasi-isometric mapping tori, is to use a result of Macura (Macura, Nataša, Detour functions and quasi-isometries., Q. J. Math. 53, No. 2, 207-239 (2002). ZBL1036.20033.).

Indeed, let $G=F\rtimes_\phi\mathbb Z$ and let $G'=H\rtimes_\phi\mathbb Z$. Suppose that $\phi$ has polynomial growth of order $r$. Then, as Lee explained above, $\phi|_H$ also has non-exponential, and thus polynomial, growth. But Macura's result (Theorem 1.1 in the above) says that $G$ has detour function a polynomial of degree exactly $r+1$, and another application of the same theorem shows that $G'$ has detour function a polynomial of degree $r'+1$, where $r'$ is the degree of the polynomial growth function of $\phi|_H$. But $G$ is obviously quasi-isometric to $G'$, and Macura shows that the order of the detour function is a quasi-isometry invariant, so $r=r'$.

(The detour function is very similar to the perhaps more familiar divergence function.)