Let $G$ be a subgroup of $S_n$, we view the elements as bijective functions of $\{1, \ldots, n\}$ to itself. Let $\chi : G \to \mathbb{C}$ be a character (group homomorphism, one-dimensional representation). Then there is a map,
$$\Delta_{G, \chi} \colon Mat(n, \mathbb{C}) \to \mathbb{C}$$
Given by:
$$\Delta_{G, \chi} ((a_{i,j})_{i, j = 1}^n) = \sum_{g \in G} \chi(g)\prod_{i=1}^n a_{i, g(i)}$$
When $G = S_n$ (the full group) there are only two choices of $\chi$: the trivial representation and the sign-representation. In these cases the definition above reproduces the standard definition of the permanent and determinant respectively.
It would seem to me that this more general class of permanent/determinant-like function has been studied before, e.g. in the 19th century when determinants were really popular. Do you know if has a name?
As per the comments, the answer is Immanant