Direct product is cancellative up to order statistics-equivalence

Statement

Suppose $G_1,G_2,H_1,H_2$ are finite groups. Suppose $G_1$ and $H_1$ are Order statistics-equivalent finite groups (?), and suppose that the External direct product (?) $G_1\times G_2$ is order statistics-equivalent to the external direct product $H_1\times H_2$. Then, $G_2$ is order statistics-equivalent to $H_2$.