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Well, in the case X is also a group, your question somehow relates with cellular automata, i.e your map $\phi :G^X\rightarrow G^X$ is continuous and X-equivariant. When X is amenable,$\phi$ is pre-injective if and only if $\phi$ is surjective. Thus, your dual $\widehat{\phi}$ will be injective in this case. It is not true anymore when $X=F_2$, the free group with 2 generators, see section 5.11 of Ceccherini-Silberstein, Coornaert's book "Cellular Automata and Groups" for an example of pre-injective but not surjective cellular automaton over $F_2$.