Integral almost square-free modular categories

We study integral almost square-free modular categories; i.e., integral modular categories of Frobenius–Perron dimension pnm, where p is a prime number, m is a square-free natural number and gcd(p,m) = 1. We prove that, if n ≤ 5 or m is prime with m < p, then they are group-theoretical. This generalizes several results in the literature and gives a partial answer to the question posed by the first author and Tucker. As an application, we prove that an integral modular category whose Frobenius–Perron dimension is odd and less than 1125 is group-theoretical.