  Let p and l be two distinct prime numbers and let G be a group. We study the
asymptotic behaviour of the mod-l Betti numbers in p-adic analytic towers of
finite index subgroups. If X is a finite l-group of automorphisms of G, our
main theorem allows to lift lower bounds for the mod-l cohomology growth in the
fixed point group G^X to lower bounds for the growth in G. We give applications
to S-arithmetic groups and we also obtain a similar result for cohomology with
rational coefficients.
