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Depending on G, you can not even check whether two homomorphisms are equal or not (due to the unsolvability of the word problem).
If you want specifically S = SymmetricGroup(4) then homomorphisms G -> S are in correspondence with the stabilizer of 1. So this is the same thing as looking at subgroups of index 4 in your group G. If your group is "nice enough" that should be faster than brute force.
