$A^* = B^*$ with $\{0,1\}$ contained in $A$ but not in $B$ I'm trying to exhibit two formal languages $A,B ⊆ \{0,1\}^*$ such that $A^* = B^*$ and $\{0,1\}$ is contained in $A$ but not in $B$.

Finding a language for $A$ is very easy, but I get stuck on $B$, because since $A^*=B^*$, and A has $\{0,1\}$, that $B^*$ must also have $\{0,1\}$. I can't think of a language that has $\{0,1\}$ in $B^*$ but would not be in $B$. Maybe I'm missing something.
If $0 \in A$ then clearly $0 \in A^* = B^*$. Conversely, if $0 \in B^*$ then $0 \in B$: indeed, if $0 = w_1 \ldots w_n \in B$, then exactly one of the $w_i$ can be non-empty, and it must equal $0$. Therefore $\{0,1\} \subseteq A$ and $A^* = B^*$ implies $\{0,1\} \subseteq B$. In other words, there are no two languages $A,B$ satisfying your constraints.