Is there a specific structure to the automorphism set of a field theory with respect to internal v. spacetime symmetries?

I'm trying to work out what it means exactly for a field to be transformed, without referring to gauges for now. As far as I can tell, from a rigorous perspective, a transformation of the field is an automorphism of the relevant vector bundle, ie, for a vector bundle

$$\pi : E \to M$$

A transformation of a section $$s \in \Gamma(E)$$ is just a member of the automorphism $$\text{Aut}(E)$$, such that the new section is, for $$\Phi \in \text{Aut}(E)$$,

$$s \to s' = \Phi(s)$$

The two big types of transformations involved in field theories are the internal ones, which seem to be vertical automorphisms, ie :

$$\Phi \in \text{Aut}(E),\ \forall x \in E,\ \pi(x) = \pi(\Phi(x))$$

and the (active) coordinate transformations. For some diffeomorphism $$\phi \in \text{Diff}(M)$$, we get some induced automorphism via the pullback

$$\Phi(s) = \phi^* s$$

Is this all correct so far, as far as the idea of a field transformation goes? Also do the two different types of transformation have some kind of structure in the global space of all bundle automorphisms? I know the automorphisms themselves induce a diffeomorphism on the manifold via

$$\begin{eqnarray} f : M &\to& M\\ \pi(x) &\mapsto& \pi(F(x)) \end{eqnarray}$$

meaning that vertical automorphisms are associated with the identity. Do the internal and coordinate automorphisms span the entire space, forming a "basis" for let's say pathological field transformations like

$$s(x) \to s'(x) = e^{i\alpha(x)} s(x + a)$$