We establish necessary and sufficient conditions for the existence of a Hermitian solution to the system of equations $A_{1}X_{1}=C_{1}, X_{1}B_{1}=D_{1}, A_{2}X_{2}=C_{2}, X_{2}B_{2}=D_{2}, A_{3}X_{1}A_{3}^{*} + A_{4}X_{2}A_{4}^{*}=C_{5}$ for adjointable operators between Hilbert $C^{*}$-modules, and provide an expression for the general Hermitian solution to the system. We present necessary and sufficient conditions for the existence of a unique Hermitian solution to the systems $A_{1}X_{1}=C_{1}, X_{1}B_{1}=D_{1}$ and $A_{3} X_{1} A_{3}^{*} = C_{5}$ of adjointable operators between Hilbert $C^{*}$-modules. Several examples are given to explain the use of the theory. Some of the findings of this paper extend some known results in the literature.

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