@ADL no, the free kernel may be infinitely generated (eg. I include surface groups in the free-by-cyclic category). And thanks for the reference! In fact, the question was motivated by this very same paper, I was trying to understand if the class of (torsion-free-virtually-free-by-cyclic) is actually strictly bigger than simply all free-by-cyclic

@Dedalus it was a rushed commentary... Actually all those statements from Shatz's book are in Serre's, although scattered in a different order. But he elaborates a bit more on his proofs, I'm reading them now to see if I get any ideas. Anyway, there is a group currenctly attacking this exercise during this week. If we make any concrete progress I'll post it here as a comment or an answer.

Check Theorem 13 (Tower Theorem) on Stephen Shatz's "Profinite Groups, Arithmetic and Geometry", page 61. Seems more general then Serre's formulation, and I think his isomorphism may solve it... but he gets away with spectral sequences though!

@ChrisGerig I haven't been so lucky, the first 5 pages of a search for "coinflation on cohomology" only shows coinflation defined at homology, not cohomology groups. Same goes for "$\text{coinf}$" references on Weibel's book, all three appearances refer to maps defined on group homology rather than cohomology.