Let *G* be a group and let *a* be a fixed element of *G*, not necessarily belongs to the center of *G*. In this paper we study the superstability of the following functional equations
$$\begin{aligned} f(ax\sigma (y))-f(axy)=2f(x)g(y),\quad x,\ y\in G, \end{aligned}$$
and
$$\begin{aligned} f(ax\sigma (y))-f(axy)=2g(x)h(y),\quad x,\ y\in G, \end{aligned}$$
where
$$\sigma : G \rightarrow G$$
is an involution and *f*, *g*, *h* are unknown complex valued functions.