I worked the proof for 3) in this link out, but I have problems with the last step:
Let $\sigma$ be a irred. representation of a normal subgroup $H=\langle z\rangle$ of $G$ and $\sigma'$ its dual representation. The proof comes to the conclusion that $\sigma \text { is equivalent to } \sigma' \Leftrightarrow z^{-1}=z^{p^k}, $ where $p$ is the charakteristic of the field $K$.
Where is the connection between saying that " $\sigma$ is equivalent to $\sigma'$ " and " $\sigma$ expands to a representation of $G$ "?