Why are these definitions of groups of central type equivalent?

Question

Let $G$ be a finite group.

1. In the celebrated paper of Howlett and Isaacs, On Groups of Central Type, Math. Z. (1983)., the group $G$ is called to be of central type if $G$ has an irreducible complex character $\chi$ such that $\chi(1)^2=[G : Z(G)]$. (In other sources, the quotient $G/Z(G)$ is called to be of central type, where $G$ satisfies above property).
   
2. For each 2-cocycle $f \in Z^2(G, \mathbb{C}^*)$, a complex twisted group algebra, denoted by $\mathbb{C}^fG$, is defined by extending the multiplication $g_1.g_2:=f(g_1, g_2)g_1g_2$, $\forall g_1, g_2 \in G$, to all linear combinations of elements of $G$ with coeffecients in $\mathbb{C}$. It is known that complex twisted group algebra $\mathbb{C}^fG$ may be a simple ring. A group $G$ is called to be of central type, if the twisted complex group algebra $\mathbb{C}^f \overline{G}$ is simple for some $2$-cocyle $f$ on $\overline{G}$, where $\overline{G}=G/Z(G)$ . (For example see the preprints of Ofir Schnabel on this topic).
   
   I know that $\mathbb{C}^fG$ is simple if and only if $G$ possesses only one class of irreducible $f$-projective character. But I can't see the relation of these two definitions. So my questions are as following:
   QUESTION 1: How simplicity of $\mathbb{C}^f \overline{G}$ implies that $G$ is of central type according to the definition given in $1$? (Update: this question is settled by the Theorem $1$ of this paper. The statement and proof of the theorem are given in answers).
   
   Further Discussions:
   
   In the paper mentioned in $1$, the authors proved the following brilliant theorem:
   
   Theorem (Howlett-Isaacs): If $N \lhd G$ and $\lambda \in Irr(N)$ is a $G$-invariant irreducible character of $N$ such that ${\lambda}^G$ is a multiple of some $\chi \in Irr(G)$, then $G/N$ is solvable.
   
   

It is mentioned in this paper that if $\mathbb{C}^fG$ is simple, then Howlett-Isaacs theorem implies that $G$ is solvable.


QUESTION 2: Why the above claim is true?

Thanks for any help.

Answer 1

The Theorem $1$ of the following paper answers our question $1$.
DeMeyer, F. R., Janusz, G.J. Finite groups with an irreducible representation of large degree. Math. Z. 108 (1969).

Answer to 2nd Question:
Assume that $\mathbb{C}^fG$ is simple. Therefore, $G$ has an irreducible complex $f$-projective character $\chi$ such that ${\chi(1)}^2=|G|$. It is well known that $\chi$ determines an irreducible ordinary character $\chi^*$ of the same degree on $G^*$, where $G^*$ is a central extension of $G$ (namely a Schur cover of $G$).
Now we have $G^*/C \cong G$ and ${\chi^*(1)}^2 = |G| = [G^* : C]$. Hence, $G^*$ is of central type (see definition 1). Restricting the character $\chi^*$ to $C$, we have ${\chi^*}_C = \chi^*(1) \lambda$, where $\lambda$ is a linear character of $C$. Now ${\lambda}^{G^*}(1) = [G^* : C]$. On the other hand, $\chi^*(1) \chi^*$ is a constituent of ${\lambda}^{G^*}$. Comparing the degrees we obtain that ${\lambda}^{G^*} =\chi^*(1) \chi^* $. Now the Howlett-Isaacs theorem yields.