Realization of the minimal extended seesaw mechanism and the $TM_2$ type neutrino mixing

We construct a neutrino mass model based on the flavour symmetry group $A_4\times C_4 \times C_6 \times C_2$ which accommodates a light sterile neutrino in the minimal extended seesaw (MES) scheme. Besides the flavour symmetry, we introduce a $U(1)$ gauge symmetry in the sterile sector and also impose CP symmetry. The vacuum alignments of the scalar fields in the model spontaneously break these symmetries and lead to the construction of the fermion mass matrices. With the help of the MES formulas, we extract the light neutrino masses and the mixing observables. In the active neutrino sector, we obtain the $\text{TM}_2$ mixing pattern with non-zero reactor angle and broken $\mu$-$\tau$ reflection symmetry. We express all the active and the sterile oscillation observables in terms of only four real model parameters. Using this highly constrained scenario we predict $\sin^2 \theta_{23} =0.545^{+0.003}_{-0.004}$, $\sin \delta = -0.911^{+0.006}_{-0.005}$, $|U_{e4}|^2 = 0.029^{+0.009}_{-0.008}$, $|U_{\mu4}|^2 = 0.010^{+0.003}_{-0.003}$ and $|U_{\tau4}|^2 = 0.006^{+0.002}_{-0.002}$ which are consistent with the current data.