Subgame Perfect Equilibrium for Pure and Mixed strategy In a game theory textbook there is something similar to the table below where there is one pure strategy nash equilibrium and multiple mixed strategy nash equilibria. It is a simultaneous game with the payoffs presented below.
If we assume that this game is played twice, How do I identify all subgame perfect equilibria for this game, as well as nash equilibrium that is not a subgame perfect equilibrium?
For a game with multiple pure strategy nash equilibrium I think I can find a solution by using backward induction, but for a game like this with just one pure strategy nash equilibrium and multiple mixed strategy nash equilibria, I have  no idea how to identify the subgame perfect equilibria and possibly a nash equilibrium that is not subgame perfect equilibrium, especially when there are mixed strategy equilibria included.
Any help in this would be appreciated.
\begin{array}{|c|c|c|c|}
\hline
  &amp;  A    &amp;  B    &amp;  C \\\hline
A &amp; (1,1) &amp; (0,0) &amp; (0,0)\\\hline
B &amp; (0,0) &amp; (2,1) &amp; (1,2)\\\hline
C &amp; (0,0) &amp; (1,2) &amp; (2,1)\\\hline
\end{array}
Check for the Nash equilibria (pure or mixed) of the one-shot game.
Repetition of the strategy profile of the Nash equilibria of the one-shot version yields one set of subgame perfect equilibria: For instance, play $(A,A)$ in the first stage and for any action profile played at the first stage, play $(A,A)$ in the second stage. The same holds true for the mixed (complete or otherwise) as well. For instance, the totally mixed SPNE is :Play $\left(\frac{3}{5},\frac{1}{5},\frac{1}{5}\right)$ in the first stage and for any action profile (assuming the randomisation is observable), play $\left(\frac{3}{5},\frac{1}{5},\frac{1}{5}\right)$ in the second stage.
Combinations of the Nash equilibria of the one-shot game yield another set of SPNE: For instance, Play $(A,A)$ in the first stage and play $\left(0,\frac{1}{2},\frac{1}{2}\right)$ for any action profile in the second stage and so on.
Use credible threats to play a non-Nash action profile in the first period: Play $(B,B)$ in the first stage. At the second stage, if $(B,B)$ was played, play $\left(0,\frac{1}{2},\frac{1}{2}\right)$. For any other action profile at the first stage, play the totally mixed Nash equilibrium of the one-shot game. Why does this work? There is no incentive for player 1 to deviate anyway because they're playing their best reply in the first stage. For player 2, if they were to deviate to $C$. they'd receive $2$ in the first stage and $1/3$ in the second stage. Assuming no discounting, they get $7/3$ while if they obeyed the strategy, they would get $(1+3/2)$ which is strictly greater. Similarly $(C,C)$, $(B,C)$ and $(C,B)$ can be played as first stage action profiles in a sub-game perfect equilibrium.