Linear maps preserving $A$-unitary operators

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