Frit is a cryptographic 384-bit permutation recently proposed by Simon et al. and follows a novel design approach for built-in countermeasures against fault attacks. We analyze the cryptanalytic security of Frit in different use cases and propose attacks on the full-round primitive. We show that the inverse \(\textsc {Frit}^{-1}\) of Frit is significantly weaker than Frit from an algebraic perspective, despite the better diffusion of the inverse of the mixing functions \(\sigma \): Its round function has an effective algebraic degree of only about 1.325. We show how to craft structured input spaces to linearize up to 4 (or, conditionally, 5) rounds and thus further reduce the degree. As a result, we propose very low-dimensional start-in-the-middle zero-sum partitioning distinguishers for unkeyed Frit, as well as integral distinguishers for reduced-round Frit and full-round \(\textsc {Frit}^{-1}\). We also consider keyed Frit variants using Even-Mansour or arbitrary round keys. By using optimized interpolation attacks and symbolically evaluating up to 5 rounds of \(\textsc {Frit}^{-1}\), we obtain key-recovery attacks with a complexity of either \(2^{59}\) chosen plaintexts and \(2^{67}\) time, or \(2^{18}\) chosen ciphertexts and time (about 5 seconds in practice).