  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
