This question may be more of a philosophical rather than mathematical nature.

Assume I have a scheme $X$ and an automorphism $F:X\longrightarrow X$ that is unramified at every closed point $\mathfrak{m}\in X$. For instance, $X$ might be of finite type over $\mathbb{F}_{p}$ and $F$ might be the Frobenius, but I do not want to restrict myself to this special case.

If $H^{*}(X,R)$ is a Weil cohomology theory (for instance $R=\mathbb{Q}_{l}$, the $l$-adic cohomology) then Grothendieck, generalizing results of Lefschetz, proved the famous trace formula \begin{equation*} \sum_{x\in X^{F}}\operatorname{tr}(F,X_{x})=\sum_{i\in\mathbb{N}}(-1)^{i+1}\operatorname{tr}(F,H^{i}(X,R))\text{.} \end{equation*}

QUESTION: Do any generalizations of this result exist in case I have two morphisms $F_{1},F_{2}$, or any finite or infinite number of automorphisms $F_{i}:X\longrightarrow X$ and $G=\left<F_{i}\mid i\in I\right>$ is the group of automorphisms of $X$ generated by the $F_{i}$'s. Is there an expression for \begin{equation*} \sum_{x\in X^{G}}\operatorname{tr}(F,X_{x})\text{?} \end{equation*}

This question may be more of a philosophical rather than mathematical nature.

Assume I have a scheme $X$ and an endomorphism $F:X\longrightarrow X$. For instance, $X$ might be of finite type over $\mathbb{F}_{p}$ and $F$ might be the Frobenius, but I do not want to restrict myself to this special case.

If $H^{*}(X,R)$ is a Weil cohomology theory (for instance $R=\mathbb{Q}_{l}$, the $l$-adic cohomology) then Grothendieck, generalizing results of Lefschetz, proved the famous trace formula \begin{equation*} \sum_{x\in X^{F}}\operatorname{tr}(F,X_{x})=\sum_{i\in\mathbb{N}}(-1)^{i+1}\operatorname{tr}(F,H^{i}(X,R))\text{.} \end{equation*}

QUESTION: Do any generalizations of this result exist in case I have two morphisms $F_{1},F_{2}$, or any finite or infinite number of automorphisms $F_{i}:X\longrightarrow X$ and $G=\left<F_{i}\mid i\in I\right>$ is the group of automorphisms of $X$ generated by the $F_{i}$'s. Is there an expression for \begin{equation*} \sum_{x\in X^{G}}\operatorname{tr}(F,X_{x})\text{?} \end{equation*}