Permute subsystems of a matrix?

Question

Suppose I have a matrix $M$ acts on the space $\mathbb C^4\otimes \mathbb C^2\otimes \mathbb C^4\otimes \mathbb C^3$. Is there a method to permute the 2nd subsystem of $M$ and the 4th subsystem of $M$? In other words, I have a matrix $M=A_1\otimes A_2\otimes A_3\otimes A_4$, how can I permute the second and the fourth subsystem of $M$ to get $A_1\otimes A_4\otimes A_3\otimes A_2$ when I only have access to $M$ while don't have access to $A_i$?

For example, I have a matrix $$M=\left( \begin{array}{cccc} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \\ \end{array} \right)$$ how can I switch the first and second subsystems permitted the first system is of dimension $2$ and the second system is of dimension $2$. In index form, $M$ can be written as $$\left( \begin{matrix} m_{00,00}& m_{00,01}& m_{00,10}& m_{00,11}\\ m_{01,00}& m_{01,01}& m_{01,10}& m_{01,11}\\ m_{10,00}& m_{10,01}& m_{10,10}& m_{10,11}\\ m_{11,00}& m_{11,01}& m_{11,10}& m_{11,11}\\ \end{matrix} \right) .$$ The element $m_{00,10}$ after switching should be in place $00,01$. Anyway, the transformed $M$ in numerical form should be $$ \left( \begin{array}{cccc} 1 & 3 & 2 & 4 \\ 9 & 11 & 10 & 12 \\ 5 & 7 & 6 & 8 \\ 13 & 15 & 14 & 16 \\ \end{array} \right). $$

Answer 1

My current solution is as follows. The core idea is to utilize the TensorTranspose function. Let's clarify things using the example in the original question. I first reshape the matrix using

testmat = ArrayReshape[mat, {2, 2, 2, 2}];

Then I switch the first and second subsystems using TensorTranspose

testmat = TensorTranspose[testmat, {2, 1, 4, 3}];

Finally, I recover the matrix from testmat using reshape function

testmat = ArrayReshape[testmat, {4, 4}];

The code above is summarized as follows

mat = ArrayReshape[Range@16, {4, 4}];
testmat = ArrayReshape[mat, {2, 2, 2, 2}];
testmat = TensorTranspose[testmat, {2, 1, 4, 3}];
testmat = ArrayReshape[testmat, {4, 4}];
testmat // MatrixForm