Let $A$ be a generic (or varying) square, real $ n \times n$ matrix. Let $G$ be a fixed $n \times k$ matrix, $k < n.$ Denote by $||.||_F$ the Frobenius norm.

Is it true that $||AG||_F \geq c(G) ||A||$, where $c(G)$ is a positive constant only depending on the matrix $G?$

Thanks in advance!