Good morning, there is a question which has arisen during my studies. It would be a pleasure to get some remarks or hints in which direction one should think to solve the problem. Let me explain it.
In the following we work over $\mathbb K=\mathbb C$ or $\mathbb R$, but will not mention it specifically. I will first give a paragraph of explanations which may be skipped.
Let $k\in\mathbb N+1 \cup \{\infty\}$. Given a $C^k$ Lie group $G$ with algebra $\mathfrak g$. If a finite dimensional vector space $V$ represents $G$ by means of a morphisms of Lie groups $\sigma: G \rightarrow \operatorname{Aut}(V)$, then he also represents $\mathfrak g$ by the derivation $d\sigma: \mathfrak g \rightarrow \operatorname{End}(V)$. Furthermore both $G$ and $\operatorname{Aut}(V)$ posess exponential maps defined by means of flows of left-invariant vector fields and denoted $\exp$ satisfying \begin{equation}\label{1}(\ast)\qquad\exp \circ d\sigma = \sigma \circ \exp\end{equation} by uniqueness of ODE solutions. To $\sigma$ we can associate the conjugation representation of $G$ by $\operatorname{End}(V)$, that is $$ Con_\sigma: G \rightarrow \operatorname{Aut} ( \operatorname{End}(V)), g \mapsto \sigma(g)\circ \cdot \circ \sigma(g^{-1})~.$$ Given a Lie algebra representation $\tau : \mathfrak g \rightarrow \operatorname{End}V$ there is an associated adjoint representation, i.e. $$ Ad_\tau: \mathfrak g \rightarrow \operatorname{End}(\operatorname{End}(V)), v \mapsto [\tau(v),\cdot]~,$$ and we have $$dCon_\sigma = Ad_{d\sigma}~.$$
Main part. Now given a $C^k$ vector bundle $E$ over a $C^k$ manifold $M$. Given a representation of $G$ by each fiber such that $G \rightarrow \operatorname{Aut}(E)$ is $C^k$. Then the derived collection of representations $\mathfrak g\rightarrow \operatorname{End}(E)$ shall be $C^k$, too. We now get representations of $G$ and $\mathfrak g$ by the sections $C^k(M,E)$. Also in this context we can define associated adjoint and conjugation representations $Ad$ and $Con$ by the probably infinite dimensional vector space $\operatorname{End}(C^k(M,E))$, respectively. We note $\exp: \operatorname{End} \operatorname{End} E \rightarrow \operatorname{Aut} \operatorname{End} E$. The question arises whether those two representations are still linked by an equation like $(\ast)$, that is $$ \exp \circ Ad = Con \circ \exp~$$ in this case.
The evident problem seems to be that endomorphisms $C^k(M,E)$ need not to be $C^k(M,\mathbb K)$-linear. What happens, for instance, if we take $E$ to be differential forms on $M$ with certain coefficients and a connection $D\in\operatorname{End}(C^k(M,E))$ on $E$?
Best regards.