Pushdown Automata (PDA) is the automaton that takes a stack as auxiliary storage.

Finite Automata (FA):
- It is the simplest form of automata that recognizes regular languages.
- It consists of a finite number of states and transitions between those states based on the input symbols.
- It does not have any memory to store information about previous inputs.

Turing Machine (TM):
- It is a more powerful computational model than FA and PDA.
- It consists of an infinite tape divided into cells, a read/write head that can move left or right on the tape, and a finite control unit.
- It can read and write symbols on the tape, change its internal state, and move the tape head.
- It has an unbounded memory, but it does not use a stack as auxiliary storage.

Pushdown Automata (PDA):
- It is an extension of FA with an added stack as auxiliary storage.
- It can recognize context-free languages, which are more powerful than regular languages.
- The stack allows PDA to have memory and keep track of previously visited states.

Why PDA takes stack as auxiliary storage?
- PDA uses a stack to keep track of the history of visited states and symbols.
- The stack allows PDA to remember information about previous inputs and make decisions based on that information.
- The stack can be used to push symbols onto it, pop symbols from it, and perform stack operations like push, pop, and peek.

Advantages of using a stack in PDA:
- Allows context-free languages to be recognized, which cannot be recognized by FA.
- Provides a way to keep track of nested structures, such as balanced parentheses or nested function calls.
- Allows PDA to simulate recursive function calls and handle recursive languages.

Conclusion:
- Finite Automata (FA) does not have any auxiliary storage.
- Pushdown Automata (PDA) uses a stack as auxiliary storage.
- Turing Machine (TM) has an unbounded memory but does not use a stack as auxiliary storage.
- Therefore, the correct answer is option 'B' - Pushdown Automata (PDA) takes a stack as auxiliary storage.