What is the number of epimorphisms from $F_3$ to $G\cong \mathbb{Z}_p\oplus\mathbb{Z}_p\oplus\mathbb{Z}_{p^5}$, for some prime $p$?
I understand that $G$ is generated by three elements (two of order $p$ and one of order $p^5$). Let's fix those generators. Each automorphism of $G$ will map them to another generator triple.
Am I correct in my assumption that there are exactly $|Aut(G)|$ epimorphisms?