A Symmetric bilinear variation for self congruence?

Question

--edited--

May I know the group or condition that satisfies $$S^T A S = A$$ possibly when $A_{n\times n}$ is symmetric?

Meanwhile, below is an example for $S_{3\times 3}$.

MATLAB code:

A = [2.9603   1.3855   1.8212    %symmetric
     1.3855   4.7366   1.2824
     1.8212   1.2824   2.653];

S = [-0.29412   -0.35451     0.71379
      0.15026    1.0698      0.18701
      1.15      -0.0036714   0.17226];

S'*A*S     % returns symmetric matrix A as 'ans' below
           % it's off a bit as I intentionally rounded the elements in the 
           % matrices above for the sake of communicating this.

% ans =
%        2.9604       1.3856       1.8212
%        1.3856       4.7367       1.2824
%        1.8212       1.2824        2.653

Thanks in advance.