Compatible systems of mod $p$ Galois representations and Hecke characters

Compatible systems of n-dimensional, mod p representations of absolute Galois groups of number fields were considered by Serre in his study of openness of images of adelic Galois representations arising from elliptic curves in [S2]. In [K] the author considered abstract compatible systems of n-dimensional, mod p representations of the absolute Galois group of Q and determined them in the one-dimensional case.