Pexider invariance equation for embeddable mean-type mappings

We prove that whenever $M_1,\dots,M_n\colon I^k \to I$, ($n,k \in \mathbb{N}$) are symmetric, continuous means on the interval $I$ and $S_1,\dots,S_m\colon I^k \to I$ ($m <n$) satisfies a sort of embeddability assumptions then for every continuous function $\mu \colon I^n \to \mathbb{R}$ which is strictly monotone in each coordinate, the functional equation $$\mu(S_1(v),\dots,S_m(v),\underbrace{F(v),\dots,F(v)}_{(n-m)\text{ times}})=\mu(M_1(v),\dots,M_n(v))$$ has the unique solution $F=F_\mu \colon I^k \to I$ which is a mean. We deliver some sufficient conditions so that $F_\mu$ is well-defined (in particular uniquely determined) and study its properties. The background of this research is to provide a broad overview of the family of Beta-type means introduced in (Himmel and Matkowski, 2018).