If G is a generalized inverse of a matrix A (i.e. AGA=A), then is it true that every generalized inverse of A can be written in the form G+B−GABAG for some matrix B of same order as G?

I will show that this matrix is a generalized inverse for every matrix B, since

$$\begin{array}{rl}A(G+B-GABAG)A& =AGA+ABA-AGABAGA\\ & =A+ABA-ABA\\ & =A\end{array}$$

But I cant conclude that every generalized inverse of A can be written in this form. Help with that

I will show that this matrix is a generalized inverse for every matrix B, since

$$\begin{array}{rl}A(G+B-GABAG)A& =AGA+ABA-AGABAGA\\ & =A+ABA-ABA\\ & =A\end{array}$$

But I cant conclude that every generalized inverse of A can be written in this form. Help with that