Where does the primary obstruction of a fibration show up in its spectral sequence?

In the general case of integral coefficients and possibly non-trivial local coefficient system, let $\pi=\pi_1(B)$. A cocycle for the obstruction class is an element $o\in Hom_{\mathbb Z\pi}(C_{k+1}\tilde{B},\pi_kF)$. Now $o$ induces a group homomorphism $$ H^0\left(B;{H^k(F)}\right)\cong Hom_{\mathbb Z}(\pi_kF,\mathbb Z)^\pi \cong Hom_{\mathbb Z\pi}(\pi_kF,\mathbb Z)\to Hom_{\mathbb Z\pi}(C_{k+1}\tilde{B},\mathbb Z).$$ Here we use the universal coefficient theorem and the fact that $H^0(B,M)=M^\pi$ for connected $B$. Since $o$ is a cocycle, the group homomorphism factors through cocycles $C^{k+1}(B;\mathbb Z)$, so, since $H^0(F)=\mathbb Z$ there is an induced map $$ H^0\left(B;{H^k(F)}\right) \to H^{k+1}(B;H^0(F))$$ which is the $d_{k+1}$-differential $$E_{2}^{0,k}= E_{k+1}^{0,k}\to E_{k+1}^{k+1,0}=E_2^{k+1,0}$$ in the spectral sequence.

Or to put it differently, there is a map $$H^{k+1}(B,\pi_k F) \to Hom(H^0(B;H^k(F)),H^{k+1}(B;H^0(F)))$$ adjoint to the composition of cup product and Hurewicz and Kronecker maps $$H^{k+1}(B,\pi_k F) \otimes H^0(B;H^k(F)) \to H^{k+1}(B;\pi_k F \otimes H^k(F))\to H^{k+1}(B;\mathbb Z)$$ which sends the obstruction class to the $d_{k+1}$-differential.

Moreover if we consider the spectral sequence $H^*(B;H^*(F;\pi_k(F)))\to H^*(E;\pi_k(F))$ the connection is even more direct: then $H^0(B;H^k(F;\pi_k(F))$ contains the distinguished element corresponding to the identity of $\pi_k F$ under the universal coefficient theorem, whose image under $d_{k+1}$ is the obstruction class in $H^{k+1}(B;H^0(F;\pi_k(F))$.

At least in the case $\pi_1(B)=0$, $\mathfrak{o}(f)$ is just the first non-trivial differential, $d_k$ in disguise (let's work over some field, for simplicity; then $d_k\in\operatorname{Hom}(H^k(F),H^{k+1}(B))\cong H^{k+1}(B)\otimes H_k(F)\ni \mathfrak o(f)$).

Reference (well, kind of: it doesn't even give precise statement, let alone proof): Mosher, Tangora. Cohomology operations and applications in homotopy theory (pp. 103, 109).

I'm afraid I can't say anything about non-simple case, though (not even sure what is the correct statement in this case).