In this talk, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying $K^2=4p_g-8$, for any even integer $p_g\geq 4$. We carry out our study by investigating the deformations of the canonical morphism $\varphi:X\to\mathbb{P}^N$ that is a quadruple Galois cover of a smooth surface of minimal degree. As a result, we show the existence of infinitely many moduli spaces, satisfying the strict Beauville inequality $p_g>2q-4$, with an irreducible component that has a proper "quadruple" sublocus where the degree of the canonical morphism jumps up. The existence of jumping subloci is in contrast with the moduli of surfaces with $K^2=2p_g-4$, studied by Horikawa. These irreducible moduli components with a jumping sublocus also present a similarity and a difference to the moduli of curves of genus $g\geq 3$, for, like in the case of curves, the degree of the canonical morphism goes down outside a closed sublocus but, unlike in the case of curves, it is never birational.