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Abstract We provide the structure of the unit group of $${\mathbb {F}}_{p^k}S_n$$ F p k S n , where $$p>n$$ p > n is a prime and $$S_n$$ S n denotes the symmetric group on n letters. We also provide the complete characterization of the unit group of the group algebra $${\mathbb {F}}_{p^k}A_6$$ F p k A 6 for $$p\ge 7$$ p ≥ 7 , where $$A_6$$ A 6 is the alternating group on 6 letters.