LAUSR

Let G be an abelian group, let $$M_{2}(\mathbb {C})$$ be the algebra of complex $$2\times 2$$ matrices, and let $$\varphi :G\rightarrow G$$ be an endomorphism that need not be involutive. We determine the solutions $$\Phi :G\rightarrow M_{2}(\mathbb {C})$$ of the matrix functional equation $$\begin{aligned} \frac{\Phi (x+y)+\Phi (x+\varphi (y))}{2}=\Phi (x)\Phi (y),\quad x,y\in G. \end{aligned}$$ This enables us to characterize the solutions $$g:G\rightarrow \mathbb {C}^{2}$$ and $$\Phi :G\rightarrow M_{2}(\mathbb {C})$$ of the following functional equation $$\begin{aligned} g(x+y)+g(x+\varphi (y))=2\Phi (y)g(x),\quad x,y\in G, \end{aligned}$$ under the invariant condition $$\Phi \circ \varphi =\Phi $$ .