quantum field theory - Wilsonian Renormalization Group and Symmetries of the EFT

I have am action $S_0$ valid up to energy scale $\Lambda_0$ with renormalisable terms.

I want to study the EFT at a lower scale $\Lambda \ll \Lambda_0$, by using the Wilsonian RG. It will give me an effective action $S^\Lambda$ with an infinite number of terms, renormalisable and not.

In which terms is this "Effective Action" different from the general effective action I can write down just by adding terms preserving the symmetry of the starting action ?

[edit:] and in particular

1) are terms generated by RG flow ALL the ones possible respecting the symmetry of the action $S_0$, or just a subset of all the (theoretically) symmetry-respecting terms? Can WRG "miss" some terms?

2) A part from anomalies, can one get a Wilsonian effective action $S^\Lambda$ with a different (enhanced or reduced) symmetry wrt the starting $S_0$ ?

With respect to these questions:

3) How is the Coleman-Weinberg potential understood in the Wilsonian RG flow ?