We consider the problem of bounding the number of permutations $σ\in S_n$ that avoid a fixed permutation $π\in S_k$ in specific indices given by a $k$-uniform hypergraph $Λ$. We obtain relatively sharp bounds in the case where $Λ$ is a random hypergraph, and find bounds in the case where $Λ$ contains many large cliques. Along the way, we prove a supersaturation version of Füredi-Hajnal, which may be of independent interest.