The identity element of A(G) is the trivial automorphism sending each element to itself and the inverse of an automorphism is its inverse as a set map.
For (b) and (c), to show that is an automorphism, you need to show first that is well-defined. Then, you need to show that is homomorphism such that . To show injectivity of , you need to show that if , then x=y, where . To show surjectivity, it suffice to show that , where .
This is some additional stuff.
Those maps in (b) is called an inner automorphism of G. If and , then , which shows . Additionally, outer automorphism can be defined as .