Commuting graph of a group action with few edges

Let $A$ be a group acting by automorphisms on the group $G\mathrm{.}$ $\mathit{\text{The}}\mathit{\text{commuting}}\mathit{\text{graph}}\mathrm{\Gamma }(G,A)\mathit{\text{of}}A\mathit{\text{-orbits}}$ of this action is the simple graph with vertex set $\{x^A:1\mathrm{\ne }x\in G\}$, the set of all $A$-orbits on $G\setminus \{1\}$, where two distinct vertices $x^A$ and $y^A$ are joined by an edge if and only if there exist $x_1\in x^A$ and $y_1\in y^A$ such that $[x_1,y_1]=1$. The present paper characterizes the groups $G$ for which $\mathrm{\Gamma }(G,A)$ is an $\mathcal{F}$-graph, that is, a connected graph which contains at most one vertex whose degree is not less than three.