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 $H_{2}$.