Twisted permutation codes, introduced recently by the second and third authors, belong to the family of frequency permutation arrays. Like some other codes in this family, such as the repetition permutation codes, they are obtained by a repetition construction applied to a smaller code (but with a "twist" allowed). The minimum distance of a twisted permutation code is known to be at least the minimum distance of a corresponding repetition permutation code, but in some instances can be larger. We construct two new infinite families of twisted permutation codes with minimum distances strictly greater than those for the corresponding repetition permutation codes. These constructions are based on two infinite families of finite groups and their representations. The first is a family of $p$-groups, for an odd prime $p$, while the second family consists of the $4$-dimensional symplectic groups over a finite field of even order. In the latter construction, properties of the graph automorphism of these symplectic groups play an important role.