We investigate the Constraint Satisfaction Problem (CSP) over templates with a group structure, and algorithms solving CSP that are equivariant, i.e. invariant under a natural group action induced by a template. Our main result is a method of proving the implication: if CSP over a coset template T is solvable by an equivariant algorithm then T is 2-Helly (or equivalently, has a majority polymorphism). Therefore bounded width, and definability in fixed-point logics, coincide with 2-Helly. Even if these facts may be derived from already known results, our new proof method has two advantages. First, the proof is short, self-contained, and completely avoids referring to the omitting-types theorems. Second, it brings to light some new connections between CSP theory and descriptive complexity theory, via a construction similar to CFI graphs.