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 ${\text{A}}^1$-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}.