PORC counting

Prove that for every n the number of isomorphism classes of perfect n-dimensional evolution algebras over $\mathbb F_q$ is a PORC function of q.

Background

Isomorphism classes are obtained from the action of $S_n\ltimes(\mathbb F_q^\times)^n$ on invertible structure matrices. Explicit formulas are known in low dimensions, but proving PORC behavior in general requires controlling the number of invertible matrices satisfying the monomial fixed-point conditions arising in Burnside's lemma.