A quotient map $X\to X/A$ that is not a Serre fibration

Question

What is an example of a CW-pair $(X,A)$ such that the quotient map $X\to X/A$, i.e. the map obtained from the pushout \begin{eqnarray} A &\to& X\\ \downarrow &&\downarrow\\ * &\to& X/A \end{eqnarray} is not a Serre fibration?

Answer 1

A Serre fibration with a path connected base has the property that all of its fibers are homotopy equivalent, and this is almost never true of a quotient map $X \to X/A$; the fiber over the basepoint is $A$ but the fiber over any other point is a point. So take, for example, $X = S^2, A = S^1$, with the map $S^1 \to S^2$ being given by some great circle.