This is all about Clifford's theorem in the complex case, which I treat in detail. The Klein $4$ subgroup that is acting on $A_{4}\times C_{\ell}$ contains three involutions: one inverts the $C_{\ell}$ but centralizes the $A_{4}$, one induces the outer automorphism of $A_{4}$, but centralizes the $C_{\ell}$, and the third inverts $C_{\ell}$ and induces the outer automorphism of $A_{4}$. You have indicated that it is the third case you are concerned with. You have not indicated whether $\ell$ is even or odd, and that makes a difference.
Now $A_{4}$ has $4$ complex irreducible characters. The trivial character and the irreducible character of degree $3$ are stable under the outer outer automorphism of $A_{4}$, so each extends in two ways to $S_{4}$. The other two irreducible characters are complex conjugates, and are interchanged by the
outer automorphism of $S_{4}$. Hence each one of them induces irreducibly to an irreducible character of $S_{4}$, and these two induced characters are equal.
( This explains the irreducible character degrees $1,1,3,3,2$ for $S_{4}$ by the way).
If $\ell$ is odd, only the trivial character of $C_{\ell}$ is stable under inversion, and this extends in two ways to a linear character of $D_{\ell}$.
For the remaining $\frac{\ell-1}{2}$ pairs of complex conjugate linear characters, we obtain one irreducible character of $D_{\ell}$ for each pair by inducing either one to $D_{\ell}$.
If $\ell$ is even, there are two real-valued linear characters of $C_{\ell}$,which are both stable under inversion, and each extends in two ways to a linear character of $D_{\ell}$.
For the remaining $\frac{\ell-2}{2}$ pairs of complex conjugate linear characters, we obtain one irreducible character of $D_{\ell}$ for each pair by inducing either one to $D_{\ell}$.
Clifford's theorem allows you to extend this analysis to $G$: Each irreducible character of $A_{4} \times C_{\ell}$ has the form $\alpha \otimes \beta,$ where $\alpha$ is an irreducible character of $A_{4}$ and $\beta$ is an irreducible character of $C_{\ell}$. The dichotomy is that $\alpha \otimes \beta$ induces irreducibly to $G$, except when $\alpha \otimes \beta$ is $G$-stable, in which case it extends in two ways to $G$ which differ only multiplication by the non-trivial linear character of $G/(A_{4} \times C_{\ell}).$
By the earlier analysis, the character $\alpha \otimes \beta$ is $G$-stable
exactly when $\alpha$ is either trivial or of degree $3$ AND $\beta$ is a real valued irreducible character of $C_{\ell}$.
Hence there are either $2$ or $4$ stable irreducible characters when $\ell$ is respectively odd or even. Each of these (half of which have degree $1$ and the other half have degree $3$) extends in two ways to $G$. There are
$\ell-1$ or $\ell-2$ non-$G$-stable irreducible characters of $A_{4} \times C_{\ell}$ when $\ell$ is respectively odd or even and these give rise to
$\frac{\ell-1}{2}$ or $\frac{\ell-2}{2}$ induced irreducible characters of degree $6$ for $G$ and the same number of irreducible characters of degree
$2$.
Working over a finite field which is not algebraically closed field $\mathbb{K}$ is slightly more complicated because Schur's Lemma takes a more complicated form. Probably the easiest way in practice is to extend $\mathbb{K}$ to a finite splitting field $\mathbb{F}$ , describe all the (absolutely) irreducible modules
for $G$ over $\mathbb{F}$, and then note that there is one irreducible $\mathbb{K}G$-module for each orbit of absolutely irreducible modules under ${\rm Gal}(\mathbb{F}/\mathbb{K})$.
The character of direct products and Clifford's Theorem are covered in most representation theory texts: Curtis and Reiner (1962) or Isaacs ( ~1970) both
treat these, though Curtis and Reiner probably has more about the case of general fields.
