All matrices are real. By $\| \cdot \|_2$ denote the matrix norm induced by $L_2$. Assume $F$ is an invertible matrix.
Consider the norm $\| X \| _\star = \| F X \| _2$.
What is the condition on $F$ so that the norm $\| \cdot \|_\star$ is sub-multiplicative?
A simple condition could be $\|F^{-1}\|_2\leq 1$ because then $$ \|XY\|_*=\|FXF^{-1}FY\|_2\leq\|FX\|_2\|F^{-1}\|_2\|FY\|_2\leq\|X\|_*\|Y\|_*. $$ It is also a necessary condition, however. Assume for simplicity that $F=\alpha I$ where $0<\alpha < 1$ so that $\|F^{-1}\|_2=\alpha^{-1}>1$, and $X=Y=I$. Then $$ \|XY\|_*=\alpha\|XY\|_2 =\alpha\not\leq \|X\|_*\|Y\|_*=\alpha^2\|X\|_2\|Y\|_2=\alpha^2. $$ In general, if $\|F^{-1}\|_2>1$ then you can find $X$ and $Y$ for which $$ \|XY\|_*\not\leq\|X\|_*\|Y\|_*. $$