Image of a crossed homomorphism in a group

If there's a homomorphism from one group to another, the image of this homomorphism is a subgroup of the group it is in. What if: there is a map which is not a homomorphism but a crossed homomorhism (ie, let F be the map from group A to B; and x,y two elements of A then F(xy)=y F(x)+F(y)) I am interested in understanding the image of this crossed homomorhism in B. Even if the image is not a subgroup of B, can I still find its size to be a divisor of the size of B.

note that: if all y in A act trivially on F(x) (for all x in A), then the crossed homomorphism becomes a homomorphism, so we can exclude this case. Also: A is an abelian group in my case.

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(Hello, Hatice.) No. Let $A$ be a group of order $2$ acting nontrivially on a group $B$ of order $3$.