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In trying to eliminate some graphs associated with character degrees of finite groups, one comes across a challenge of not having enough tools to do the same. This raises a question on whether one can construct a group with favourable outcome. For example:

Suppose I need to construct a group $G$ with $G/N$ almost simple, where $N\unlhd G$ is the solvable socle. Suppose $\theta$ is a character of $N$ that has some degree and I want $G$ to have an irreducible constituent of $\theta^G$ in $G$. I want $I_G(\theta)$ to have some index.

How can I construct this group or rather show that it does not exist?

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