The determinant hom-counting conjecture for finite groups

Let $G_1$ and $G_2$ be finite groups, and define

$$a_{i,j}=h_{G_i}(G_j)$$

for $i,j=1,2$, where $h_G(H)$ denotes the number of group homomorphisms from $G$ to $H$.

Determinant hom-counting conjecture. The following are equivalent:

1. There is an isomorphism $G_1\cong G_2$.

$$\det\big((a_{i,j})\big)=0.$$

This is presented as a more optimistic conjecture than the preceding two-test hom-counting claim. The supplied text proves the result in the special case where $G_2$ is abelian of square-free exponent, but does not establish the general equivalence.