A transfer morphism for Hermitian K-theory of schemes with involution

In this paper, we consider the Hermitian $K$-theory of schemes with involution, for which we construct a transfer morphism and prove a version of the d\'{e}vissage theorem. This theorem is then used to compute the Hermitian $K$-theory of $\mathbb{P}^1$ with involution given by $[X:Y] \mapsto [Y:X]$. We also prove the $C_2$-equivariant $\A^1$-invariance of Hermitian $K$-theory, which confirms the representability of Hermitian $K$-theory in the $C_2$-equivariant motivic homotopy category of Heller, Krishna and \{O}stv\ae r \cite{HKO14}