Five-lemma for the end of long exact sequences of homotopy groups

This is covered in Exercise 9 in Section 4.1 of Allen Hatcher's book (page 358). It turns out that if (for all choices of base-points) $f_1,f_2,f_4,$ and $f_5$ are all bijections then $f_3$ is a bijection. Note that you can also extend one more term to the right in your diagrams, so that $\pi_0(X) \rightarrow \pi_0(X,A) \rightarrow 0$ if you define $\pi_0(X,A,x_0) = \pi_0(X,x_0)/\pi_0(A,x_0)$

The abstract case is handled in

R. Brown, ``Fibrations of groupoids'', J. Algebra 15 (1970) 103-132.

where a 5-lemma type result is Theorem 4.9. I find it easier to handle the abstract case rather than the topological example. You get injectivity on Ker( $\pi_1(X, A) \to \pi_0 A)$ and an Example 4.10 (for groupoids) shows you do not always get injectivity.

There are other applications of the exact sequences of a fibration of groupoids, see

Heath, P.R. and Kamps, K.H. "On exact orbit sequences", Illinois J. Math. 26~(1) (1982) 149--154.

One aim of all editions (from 1968) of the book Topology and Groupoids is to work in the algebraic context of groupoids where appropriate, to see which results are really on groupoids.