Upper bound on $F(X_1,X_2) = \|-2 (X_1-X_2)A + B(X_1-X_2)\|_F $ I would like to find an upper bound on
$$F(X_1,X_2) := \left\| -2 (X_1-X_2)A + B(X_1-X_2) \right\|_F$$
where $X_1, X_2 \in \mathbb R^{n \times m}$, matrices $A \in \mathbb R^{m \times m}$ and $B \in \mathbb R^{n \times n}$ are given, and $\| \cdot \|_F$ denotes the Frobenius matrix norm. My goal is to find
$$F(X_1,X_2) = \left\| -2 (X_1-X_2)A + B(X_1-X_2) \right\|_F  \le  L \| X_1 - X_2 \|_F$$
where $L$ depends on $A$ and $B$. Does such an $L$ exist? Thanks.
 A: Let $\rm Y := X_1 - X_2$. Using vectorization,
$$\begin{array}{rl} \| -2 \mathrm Y \mathrm A + \mathrm B \mathrm Y \|_{\text{F}}^2 &= \| \mbox{vec} \left( -2 \mathrm Y \mathrm A + \mathrm B \mathrm Y \right) \|_2^2\\ &= \| \mbox{vec} \left( -2 \mathrm I_n \mathrm Y \mathrm A + \mathrm B \mathrm Y \mathrm I_m \right) \|_2^2\\ &= \| -2 (\mathrm A^\top \otimes \mathrm I_n) \,\mbox{vec} \left( \mathrm Y \right)  + (\mathrm I_m \otimes \mathrm B) \,\mbox{vec} \left( \mathrm Y \right) \|_2^2\\ &= \left\| \left( (\mathrm I_m \otimes \mathrm B) -2 (\mathrm A^\top \otimes \mathrm I_n) \right) \,\mbox{vec} \left( \mathrm Y \right) \right\|_2^2\\ &\leq \left\| (\mathrm I_m \otimes \mathrm B) -2 (\mathrm A^\top \otimes \mathrm I_n) \right\|_2^2 \, \left\| \mbox{vec} \left( \mathrm Y \right) \right\|_2^2\\ &= \left\| (\mathrm I_m \otimes \mathrm B) - 2 (\mathrm A^\top \otimes \mathrm I_n) \right\|_2^2 \, \| \mathrm Y \|_{\text{F}}^2\end{array}$$
Thus, one desired constant would be the following spectral norm
$$L := \color{blue}{\left\| (\mathrm I_m \otimes \mathrm B) -2 (\mathrm A^\top \otimes \mathrm I_n) \right\|_2}$$
Alternatively, using the triangle inequality and the submultiplicativity of the Frobenius norm,
$$\begin{array}{rl} \| -2 \mathrm Y \mathrm A + \mathrm B \mathrm Y \|_{\text{F}} &\leq \| -2 \mathrm Y \mathrm A \|_{\text{F}} + \| \mathrm B \mathrm Y \|_{\text{F}}\\ &= 2 \| \mathrm Y \mathrm A \|_{\text{F}} + \| \mathrm B \mathrm Y \|_{\text{F}}\\ &\leq 2 \| \mathrm Y \|_{\text{F}} \, \| \mathrm A \|_2  + \| \mathrm B \|_2 \, \| \mathrm Y \|_{\text{F}}\\ &= \left( 2 \| \mathrm A \|_2  + \| \mathrm B \|_2 \right)  \| \mathrm Y \|_{\text{F}}\end{array}$$
and, thus, another possible constant would be the following
$$L := \color{blue}{2 \| \mathrm A \|_2  + \| \mathrm B \|_2}$$

Addendum
Using the triangle inequality and spectral properties of the Kronecker product,
$$\left\| (\mathrm I_m \otimes \mathrm B) -2 (\mathrm A^\top \otimes \mathrm I_n) \right\|_2 \leq \left\| \mathrm I_m \otimes \mathrm B \right\|_2 + 2 \left\| \mathrm A^\top \otimes \mathrm I_n \right\|_2 = \left\| \mathrm B \right\|_2 + 2 \left\| \mathrm A^\top \right\|_2 = 2 \left\| \mathrm A \right\|_2 + \left\| \mathrm B \right\|_2$$
