Frit is a new lightweight 384-bit cryptographic permutation proposed by Simon et al., which is designed for resisting fault injection and performs competitively in both hardware and software. Dobraunig et al. first studied Frit in EM construction, and left an open problem to explore the security of Frit in a sponge or duplex modes. In this paper, by introducing a new key-dependent cube attack method, we partially answer the open question by Dobraunig et al. and give some key-recovery attacks on the rounded-reduced Frit used in duplex authenticated encryption mode (Frit-AE). Our results cover all the versions of Frit-AE and include some practical key-recovery attacks that could recover the key within several minutes.