On rainbow caterpillars in elementary $p$-groups

Given a finite Abelian group $(A,+)$, consider a tree $T$ with $\mathrm{\mid }A\mathrm{\mid }$ vertices. The labeling $f\text{\hspace{0.17em}⁣}:V(T)\to A$ 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.