On Galois equivariance of homomorphisms between torsion potentially crystalline representations

Let K be a complete discrete valuation field of mixed characteristic (0,p) with perfect residue field. Let (\pi_n)_{n\ge 0} be a system of p-power roots of a uniformizer \pi=\pi_0 of K with \pi^p_{n+1}=\pi_n, and define G_s (resp.\ G_{\infty}) the absolute Galois group of K(\pi_s) (resp.\ K_{\infty}:=\bigcup_{n\ge 0} K(\pi_n)). In this paper, we study G_s-equivatiantness properties of G_{\infty}-equivariant homomorphisms between torsion (potentially) crystalline representations