My basic understanding is that a field theory consists of symmetry groups, a space $S$ that the symmetry groups act on and of fields defined on that space $S$. In other words, the space $S$ is the domain of the fields. I have understood that the space $S$ can be a rather complex construct, usually a tensor product of the Minkowski space and other spaces that the gauge groups act on.
We define the Lagrangian as a composition of the fields and require it to be invariant under the symmetry group transformations of the domain space $S$.
Then, for example, a Lorentz transformation turns the scalar field $\phi(x_\mu)$ into $\phi(L_\mu^\nu x^\mu)$. The codomain, the space of field values, is part of $S$, it has no symmetry groups defined on it and it is not being transformed.
I would then expect that a field $A^\nu(x_\mu)$ is transformed into $A^\nu(L^\alpha_\mu x_\alpha)$.
However, applying a the Lorentz Symmetry transformation to, e.g. the Lagrangian of QED requires me to do something like: $L^\nu_\beta A^\beta(L^\alpha_\mu x_\alpha)$.
Now, suddenly, the codomain of the fields is subject to transformations just like the field domain $S$.
It has always confused me that the codomain of the fields is also transformed. I thought the symmetry groups are defined on the domain of the fields?
Can it maybe be seen in a way that only the field domain is being transformed? Maybe we can re-define the vector fields a scalar fields on a bigger domain?
I will be very grateful is someone can finally clear this up for me.