Generalized inverse of a matrix on new columns

Question

Assume that there exists a matrix $A \in R^{m \times n}(m\neq n)$ whose generalized inverse matrix is $X$, and $X$ satifies the formula: $$ A=AXA\\ (AX)^T=AX $$ How to calculate the generalized inverse matrix $Y$ of $B=\left[ A,a \right] \left(a\in R^{m\times t},m\neq n\right)$,which also satifies the formula: $$ B=BYB\\ (BY)^T=BY $$