Given a finite Abelian group $(A,+)$, consider a tree $T$ with $|A|$ vertices. The labeling $f \colon V (T) \rightarrow A$ of the vertices of some graph $G$ induces an edge labeling in $G$, thus the edge $uv$ receives the label $f (u) + f (v)$. The tree $T$ is $A$-rainbow colored if $f$ is a bijection and edges have different colors. In this paper, we give necessary and sufficient conditions for a caterpillar with three spine vertices to be $A$-rainbow, when $A$ is an elementary $p$-group.