Summary: Let \(\mathcal{H}\) be a complex Hilbert space, \(A\) a positive operator with closed range in \(\mathcal{B}(\mathcal{H})\) and \(\mathcal{B}_{A}(\mathcal{H})\) the sub-algebra of \(\mathcal{B}(\mathcal{H})\) of all \(A\)-self-adjoint operators. Assume \(\varphi\colon\mathcal{B}_{A}(\mathcal{H})\) onto itself is a linear continuous map. This paper shows that if \(\varphi\) preserves \(A\)-unitary operators such that \(\varphi(I)=P\) then \(\psi\) defined by \(\psi(T)=P\varphi(PT)\) is a homomorphism or an anti-homomorphism and \(\psi(T^{\sharp})=\psi(T)^{\sharp}\) for all \(T\in\mathcal{B}_{A}(\mathcal{H})\), where \(P=A^+A\) and \(A^+\) is the Moore-Penrose inverse of \(A\). A similar result is also true if \(\varphi\) preserves \(A\)-quasi-unitary operators in both directions such that there exists an operator \(T\) satisfying \(P\varphi(T)=P\).