Variations on average character degrees and solvability

Let $G$ be a finite group, $\mathbb{F}$ be one of the fields $\mathbb{Q},\mathbb{R}$ or $\mathbb{C}$, and $N$ be a non-trivial normal subgroup of $G$. Let ${\mathrm{a}\mathrm{c}\mathrm{d}}_{\mathbb{F}}^{\ast }(G)$ and ${\mathrm{a}\mathrm{c}\mathrm{d}}_{\mathbb{F},even}(G\mathrm{\mid }N)$ be the average degree of all non-linear $\mathbb{F}$-valued irreducible characters of $G$ and of even degree $\mathbb{F}$-valued irreducible characters of $G$ whose kernels do not contain $N$, respectively. We assume the average of an empty set is $0$ for more convenience. In this paper we prove that if ${\mathrm{a}\mathrm{c}\mathrm{d}}_{\mathbb{Q}}^{\ast }(G)<9\mathrm{/}2$ or $0<{\mathrm{a}\mathrm{c}\mathrm{d}}_{\mathbb{Q},even}(G\mathrm{\mid }N)<4$, then $G$ is solvable. Moreover, setting $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$, we obtain the solvability of $G$ by assuming ${\mathrm{a}\mathrm{c}\mathrm{d}}_{\mathbb{F}}^{\ast }(G)<29\mathrm{/}8$ or $0<{\mathrm{a}\mathrm{c}\mathrm{d}}_{\mathbb{F},even}(G\mathrm{\mid }N)<7\mathrm{/}2$, and we conclude the solvability of $N$ when $0<{\mathrm{a}\mathrm{c}\mathrm{d}}_{\mathbb{F},even}(G\mathrm{\mid }N)<18\mathrm{/}5$. Replacing $N$ by $G$ in ${\mathrm{a}\mathrm{c}\mathrm{d}}_{\mathbb{F},even}(G\mathrm{\mid }N)$ gives us an extended form of a result by Moreto and Nguyen. Examples are given to show that all the bounds are sharp.