We give first an easy construction of surfaces with \(p_g=q=2, K^2=5\) and Albanese map of degree 3, describing a unirational irreducible connected component of the moduli space of surfaces of general type, which we show to be the only one of the Main Stream fulfilling the Gorenstein Assumption (see Assumption 1.2) with these invariants. We call it the family of CHPP surfaces, since it contains the family constructed by Chen and Hacon [Pac. J. Math. 223, No. 2, 219–228 (2006)] and coincides with the one considered by Penegini and Polizzi [Osaka J. Math. 50, No. 3, 643–686 (2013)]. We also give an easy construction of an irreducible connected component of the moduli space of surfaces of general type with \(p_g=q=2, K^2=6\) , and Albanese map of degree 4, which we call the family of PP4 surfaces since it contains the family constructed in Penegini and Polizzi [J. London Math. Soc. 90, No. 3, 741–762 (2014)]. Finally, we answer a question posed by Hacon and Chen [Pac. J. Math. 223, No. 2, 219–228 (2006)], via three families of surfaces with \(p_g=q\) whose Tschirnhaus module has a kernel realization with quotient a nontrivial homogeneous bundle. Two families have \(p_g=q=3\) , and the third is a new family of surfaces with \(p_g=q=2, K^2=6\) , and Albanese map of degree 3.