Ltac inv_tac :=
apply: esym (etrans _ (mul1g _)); apply: canRL (mulgK _) _;
let a := fresh "a" in apply/permP => a;
apply/eqP; rewrite permM !permE; case: a; do 4?case.

Lemma burnside_app2 : (square_coloring_number2 * 2 = n ^ 4 + n ^ 2)%N.