For a more elementary approach, avoiding induction and Mackey theory, you might try a concrete construction. Realize Aakumadula's group $G$ as a $2 \times 2$ matrix group over $\mathbb{F}_p$ (say for $p$ an odd prime) consisting of all $\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix}$. Here $a, b$ run over respectively the multiplicative group and the additive group of the field. This realizes a semidirect product $A \ltimes B$ having normal Sylow $p$-subgroup $B$ consisting of matrices with $a=1$, acted on by its (cyclic) automorphism group $A$ of order $p-1$ (the diagonal group acting by conjugation).

Check first that the commutator subgroup is just $B$, so its index $p-1$ in $G$ counts the number of *linear characters* (those complex irreducible characters of degree 1). Since the sum of squares of degrees adds up to the group order $p(p-1)$, the problem is to see that there is only one more irreducible character (of degree necessarily $p-1$). As Aakumadula suggests, you might construct this directly by induction from a nontrivial linear character of $B$. (But then you'd have to check irreducibility.)

On the other hand, another very classical fact is that the number of (distinct) irreducible characters equals the number of *conjugacy classes* in $G$. By linear algebra, conjugates must have the same eigenvalues. Sylow theory shows that elements of order $p$ (with both eigenvalues 1) are all in the normal subgroup $B$, and by conjugation with $A$ these $p-1$ elements are all conjugate. Along with the trivial class you have so far just 2 classes. But then it's easy to check that each of the $p-2$ elements with fixed $a \neq 1$ is conjugate in $G$ to precisely the $p$ elements sharing the same eigenvalues $a, 1$. Now you have $p$ classes, with no more possible eigenvalues (or group elements) to consider.

P.S. As the answers (and comment about arbitrary finite fields) indicate, the question can be approached narrowly or more broadly. What works best depends heavily on what one already knows. The approach I've sketched is deliberately elementary, restricted to the most basic knowledge of character theory, linear algebra, groups and rings. Even here there are lots of shortcuts and variants. But what is the motivation other than curiosity?