The two-test hom-counting conjecture for finite groups

Let $G_1$ and $G_2$ be finite groups. For a finite group $G$, write $h_G(H)$ for the number of group homomorphisms from $G$ to $H$, and $h^G(H)$ for the number of group homomorphisms from $H$ to $G$.

Two-test hom-counting conjecture. The following are equivalent:

1. There is an isomorphism $G_1\cong G_2$.
2. $h_{G_1}(G_i)=h_{G_2}(G_i)$ for $i=1,2$.
3. $h^{G_1}(G_i)=h^{G_2}(G_i)$ for $i=1,2$.

The conjecture proposes that testing homomorphism counts only on $G_1$ and $G_2$ suffices to distinguish any two non-isomorphic finite groups, in both covariant and contravariant directions; the preceding results establish existence of a finite distinguishing test group but do not identify one of these two groups as sufficient.