Class function

Symbol-free definition
A class function' on a group is defined as a function (to any set) that takes the same value on any two conjugate elements. Equivalently, it is a function on the group that is constant on conjugacy classes, and hence descends to a function from the set of conjugacy classes.

Definition with symbols
A class function on a group $$G$$ is a function $$f$$ from $$G$$ to some set $$X$$ such that $$f(g) = f(hgh^{-1})$$ for any $$g,h \in G$$.

Conjugacy classes of images are class functions
Let $$\rho:G \to H$$ be a homomorphism. Then the function that sends each $$g$$ to the conjugacy class of $$\rho(g)$$ is a class function. This follows from the fact that if two elements in $$G$$ are conjugate, their images in $$H$$ are also conjugate.

Characters of linear representations are class functions
For any linear representation, the character of that linear representation, viz the map that sends each group element to the trace of the corresponding linear operator, is a class function. This follows from the fact that the character depends only on the conjugacy class of the linear operator corresponding to the group element.