Let $\mathbf G$ be a connected reductive group defined over a finite field with $q$ elements and let $F : {\mathbf G} \to {\mathbf G}$ be the corresponding Frobenius endomorphism. We prove that the Mackey formula for Lusztig induction and restriction holds in ${\mathbf G}^F$ if all the simple components of ${\mathbf G}$ are of type A. Our result holds for every Frobenius endomorphism (split or non-split) and for every $q$. 1991 Mathematics Subject Classification: 20G05, 20G40.