Invariant subrings of the induced ring on the $4\times4$ symplectic group

It is proved that if $\Omega$ is an invariant subring of the induced ring $\operatorname{Ind}_{\mathrm{Sp}_4}^{\varphi,P}(K)$ with ring of values $A$ and maximal induced subring $\operatorname{Ind}_{\mathrm{Sp}_4}^{\varphi,P}(K)$, then
$$ 0\to\Omega/\operatorname{Ind}_{\mathrm{Sp}_4}^{\varphi,P}(I)\to\operatorname{Ind}_{GL_2}^{\varphi,B}(A/I) \quad\text{and}\quad 0\to A/I\to\operatorname{Ind}_{GL_2}^{\varphi,B}(A/\mathscr F), $$
and the ideals $~I$ and $\mathscr F$ of $A$ are described.
Bibliography: 2 titles.