Set of fixed points of a collection of composed functions

Suppose $h:S^nrightarrow S^n$ is a continuous function from an $n$-dimensional simplex in itself. Besides, assume that the set $Sigma$ of its fixed points is composed by a finite number of connected components.
Now consider any collection $G$ of finitely many continuous functions $g_i:S^nrightarrow S^n $ indexed by $iin I$. Assume that any two functions in $G$ are homotopic.
Create a new collection $tilde{G}$ of continuous functions $hcirc g_i$.
Assume that any fixed point of a function $hcirc g_i$ is also a fixed point of $h$. Besides given a fixed point $sigma$ of $h$, there exist a function $g_i$ in $G$ such that $sigma$ is also a fixed point of $hcirc g_i$.
Does exist one (connected) component of $Sigma$ that includes at least one fixed point of every function $hcirc g_i$?