\chi_2$, it is easy to see $\delta(\rho_1, \rho_2) = \delta(\tau_1, \tau_2)$.

In the simplest situation, this method gives the following.

\[cor:method\] Let $\mathcal H$ be the set of minimal lifts of $\bar G_1$ and $\bar G_2$ to $\operatorname{GL}_n({\mathbb C})$. Suppose that $H$ has no $Z(H)$-free normal subgroups for all $H \in \mathcal H$. Then $$\delta(\rho_1, \rho_2) \ge \min  \{ \frac 12 \alpha_n(H), \delta^\natural_n(H) : H \in \mathcal H \}.$$

This corollary will address most but not all cases of our proof of Theorem \[thm1\]. Namely, when $n=3$, it can happen that $\bar G_1$ has a lift $H \simeq \bar G_1$ which is simple, so $H$ is a $Z(H)$-free normal subgroup of itself. So we will need to augment this approach when $H_1$ or $H_2$ is simple.

Primitive degree 2 characters
=============================

In this section we will prove the $n=2$ case of Theorem \[thm1\].

We used the computer package GAP 4 [@gap] for explicit group and character calculations in this section and the next. We use the notation $[n, m]$ for the $m$-th group of order $n$ in the Small Groups Library, which is accessible by the command `SmallGroup(n,m)` in GAP. We can enumerate all (central or not) extensions of $G$ 