How to write a matrix in a different basis by using Mathematica

I have these two 4×4 matrix

Subscript(S, 1) = KroneckerProduct(PauliMatrix(1), IdentityMatrix(2))
Subscript(S, 2) = KroneckerProduct(IdentityMatrix(2), PauliMatrix(2))

Which are written in the Canonical basis ({{{1}, {0}, {0}, {0}};{{{0}, {1}, {0}, {0}}; {{{0}, {0}, {1}, {0}}; {{{0}, {0}, {0}, {1}})

I would like to write down these two matrices in the following basis:

G := {{0}, {0}, {0}, {1}}

S := (1/Sqrt(2)) {{0}, {1}, {1}, {0}}

A := (1/Sqrt(2)) {{0}, {1}, {-1}, {0}}

EE := {{1}, {0}, {0}, {0}}

Is there any way to do it in Mathematica?

Thanks in advance