If looking at one of the sub-games the player is indifferent between two actions. How does the backward induction work to recognize sub-game perfection?
I.e., suppose player 3 has two action $A_3 = \{L,M\}$ and at a specific sub-game the utility of both actions are the same $u_3(L,a_{-3}) = u_3(M,a_{-3})$. How can we proceed to determine whether a sub-game perfection exists?
You can break the "tie" between the two actions in any way you want. In your example, you will have at least two different subgame perfect equilibria. If the game is finite, you will always can find all the subgame perfect equilibria by backwards induction, the only difference is that there will be more than one way to "solve" the game.