how to extend a map to a vector space over a finite field (?)

Consider the group G := (Z/2)^5, viewed as a vector space over a finite field; and suppose we've defined a map f on the 'standard' basis elements (1, 0, 0, 0, 0), ..., (0, 0, 0, 0, 1), taking values in some finite group H.

Under what conditions will this map actually extend to
a homomorphism on all of G?

The only real stipulation here that's evident
is that these elements of G must be mapped under f to elements of order 1 or 2 in the target group H.

Is this actually enough to ensure that f extends
to a homomorphism G -> H?
.

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