I'm a little confused about step 3 of the proof of the main theorem (page 8) of the paper Limit Points in the Range of the Commuting Probability Function on Finite Groups by Peter Hegarty.
He considers a finite group $G$ and an irreducible representation $\varphi\colon G\to GL(2,\mathbb{C})$ of degree 2 (he also assumes that $Z(G)\leq G^\prime$ but I don't think that this is relevant).
There is a quotient map $\pi\colon GL(2,\mathbb{C})\to PGL(2,\mathbb{C})$. Set $K=\ker\varphi$ and $L=\ker(\pi\circ\varphi)$. Clearly $K\leq L$. He claims that $G/L\cong\text{im}(\pi\circ\varphi)$ cannot be cyclic. I believe that $G/K\cong\text{im}(\varphi)$ cannot be cyclic as otherwise $\varphi$ would not be irreducible. This doesn't explain why the smaller group $G/L$ cannot be cyclic.
Why can't $G/L$ be cyclic?