In the present paper, we prove that an open continuous homomorphism between the absolute Galois groups of p-adic local fields is geometric [i.e., roughly speaking, arises from an embedding of fields] if and only if the homomorphism is HT-preserving [i.e., roughly speaking, satisfies the condition that the pull-back by the homomorphism of every Hodge-Tate representation is Hodge-Tate].
"A note on the geometricity of open homomorphisms between the absolute Galois groups of p-adic local fields." Kodai Math. J. 36 (2) 284 - 298, June 2013. https://doi.org/10.2996/kmj/1372337519