We investigate the unsteady heat (energy) transport in an infinite mass-in-mass chain with a given initial temperature profile. The chain consists of two sublattices: the β-Fermi-Pasta-Ulam-Tsingou (FPUT) chain and oscillators (of a different mass) connected to each FPUT particle. Initial conditions are such that initial kinetic temperatures of the FPUT particles and the oscillators are equal. Using the harmonic theory, we analytically describe evolution of these two temperatures in the ballistic regime. In particular, we derive a closed-form fundamental solution and solution for a sinusoidal initial temperature profile in the case when the oscillators are significantly lighter than the FPUT particles. The harmonic theory predicts that during the heat transfer the temperatures of sublattices are significantly different, while initially and finally (at large times) they are equal. This may look like an artifact of the harmonic approximation, but we show that it is not the case. Two distinct temperatures are also observed in the anharmonic case, even when the heat transport regime is no longer quasiballistic. We show that the value of the nonlinearity coefficient required to equalize the temperatures strongly depends on the particle mass ratio. If the oscillators are much lighter than the FPUT particles, then a fairly strong nonlinearity is required for the equalization.