The question is in the title, a rephrasing could be is any finite group representable as the automorphism group of a finite tree, if not what is typically unrepresentable?
In case of ambiguity :
An homomorphism of finite rooted trees must preserved the root, and so does an isomorphism which is called an automorphism.
The cause/spirit of the question is : I make the isomorphism class of finite graphs smaller by specifying a group acting on the graph's vertices, that is an isomorphism must respect the group action (instead of the bigger $S(n)$ action).
Do I loose something by restricting myself to trees automorphism instead of group?