  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}.
