Let $\mathcal{L}(\cdot,\cdot\cdot)$ be the vector space of linear transformations on $\cdot$ to $\cdot\cdot$.
In Eaton's 2007 Multivariate Statistics, the following definition of the Kronecker product appears:
Let $(V_1,(\cdot,\cdot)_1)$, $(V_2,(\cdot,\cdot)_2)$, $(W_1,[\cdot,\cdot]_1)$, $(W_1,[\cdot,\cdot]_1)$ be inner product spaces. For $A\in \mathcal{L}(V_1,V_2)$ and $B\in \mathcal{L}(W_1,W_2)$, the Kroncker product of $B$ and $A$, denoted by $B\otimes_K A$, is a linear transformation on $\mathcal{L}(V_1,W_1)$ to $\mathcal{L}(V_2,W_2)$, defined by $$ (B\otimes_K A)C\equiv BCA' $$ for all $C\in\mathcal{L}(V_1,W_1)$. (The author adds: by $A'$, we mean the linear tranformation on $V_2$ to $V_1$, which satisfies $(x_2,Ax_1)_2=(A'x_2,x_1)_1$ for $x_1\in V_1$ and $x_2\in V_2$).
I'm adding the subscript $K$ to distinguish the above from the more pedestrian $\otimes$ defined here? In particular, if $A=A_{ij}$ and $B=B_{st}$ are real matrices, what is $A\otimes_K B$, if written as a matrix, in terms of the entries of $A$ and $B$?
From the same page we see the formula
$$ (\mathbf{B}^{\rm T} \otimes \mathbf{A})\operatorname{vec}(\mathbf{X}) = \operatorname{vec}(\mathbf{A}\mathbf{X}\mathbf{B}). $$
Where $\otimes$ is the "usual" Kronecker product as defined on the Wikipedia entry. This means the map $\operatorname{vec}(\mathbf{X}) \mapsto \operatorname{vec}(\mathbf{A}\mathbf{X}\mathbf{B})$ has the matrix form $(\mathbf{B}^{\rm T} \otimes \mathbf{A})$.
Therefore, your map, $C \mapsto BCA'$, after vectorizing, takes the form $A \otimes B$.