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`.