On solutions of the second generalization of d'Alembert's functional equation on a restricted domain

Let A be a subgroup of an abelian group (G,+) and P be a quadratically closed field with char P≠2. We give a full description of all pairs of functions f:G→P,g:A→P satisfying the equation(a)f(x+y)+f(x-y)=2g(x)f(y)(x,y)∈A×G.We present an example of solution (f,g) of (a) that cannot be extended to a solution (f,g¯) of the equation (b)f(x+y)+f(x-y)=2g¯(x)f(y)x,y∈G.