Let $H$ be a finite index subgroup of a finitely generated group $G$. Assume that $Out(H)$ is finite. Can $Out(G)$ be infinite?

$\begingroup$ followup: mathoverflow.net/questions/206585/… $\endgroup$ – YCor Jul 2 '15 at 20:25
I think the answer is yes. Let $A$ be a f.g. group. Assume that $\text{Out}(A)=1$ ($\text{Out}(A)$ finite would probably be enough) and that $A$ contains a central copy of $C^{(\infty)}$, the direct sum of countably many copies of the cyclic group $C$ of prime order $p$. Then $\text{Out}(A\times C)$ is infinite: indeed if $f$ is a homomorphism from $C$ to the center of $A$, then if we set $u_f(a,t)=(af(t),t)$, then $f\mapsto u_f$ injects $\text{Hom}(C,Z(A))$ into $\text{Out}(A\times C)$. Actually all these automorphisms are identity on the finite index subgroup $A$, and this is essentially the reason I don't expect Mark's argument to work.
It remains to find an example of such a group $A$. I expect some complicated nilpotentbyabelian group over $\mathbf{F}_p[t]$ to work (or KacMoody groups??), but I'm not prone to go into computations and maybe somebody else has one example.
Edit: Anton Klyachko's answer to my subsequent question confirms the existence of such a group $A$. Therefore, the answer to the question here is positive:
for finitely generated groups, having infinite outer automorphism group does not pass to finite index subgroups.

$\begingroup$ There is a nice result by Ould Houcine: every countable abelian group injects into the centre of some finitely presented group: math.univlyon1.fr/~ould/Fichespapiers/embeddings.pdf $\endgroup$ – Alain Valette May 1 '11 at 20:47

$\begingroup$ Yes but there are many classical ways to inject an infinitedimensional $\mathbf{F}_p$vector space into the center of a f.p. group (e.g. into Abels' group over $\mathbf{F}_p$). But we need it to have finite Out. I think it should work by picking some suitable f.g. $\mathbf{F}_p$algebra $A$ with a finite automorphism group, with an infinite $K_2$. Probably some localization of $\mathbf{F}_p[u,v]$ should work. Then the universal central extension of $\mathbf{SL}_d(A)$ should be the example. YC $\endgroup$ – YCor May 2 '11 at 11:34