"The proof we present for finitely generated free Z_p-modules is due to Gergely Zábrádi,
see ([Z10], Lemma 4.1 and Proposition 4.2), who considered special groups H and G
(in fact he considered the GL2-case for elliptic curves without complex multiplication),
but this proof can be adapted to work in the setting described in section 1.2.1. In
contrast to Zábrádi, we then show that the general case of any finitely generated
Z_p-module, not necessarily free, follows from the case of free modules.
Ulrich Schmitt. Towards a Twist Conjecture in Non-Commutative Iwasawa Theory. (2014)