Power set of $A$ consists of all subsets of $A$ and from the definition of a subset, $\phi$ is a subset of any set. So, $I$ and $II$ are TRUE.
$5$ and $\{6\}$ are elements of $A$ and hence $\{5, \{6\} \}$ is a subset of $A$ and hence an element of $2^{A}$. An element of a set is never a subset of the set. For that the element must be inside a set- i.e., a singleton set containing the element is a subset of the set, but the element itself is not. Here, option $IV$ is false. To make IV true we have to do as follows:
$\{5, \{6\} \}$ is an element of $2^{A}$. So, $\{ \{5, \{6\} \}\}\subseteq 2^{A}.$
So, option C.