In general, if $G$ is an algebraic group and $H$ is a subgroup for which $G/H$ is defined, both defined over a field $k$, $G(k)/H(k)$ need not be equal to $(G/H)(k)$. This is explained here: Quotient of group schemes and its rational points.
How would this work for $G=GL_2$ and $H=Z(G)$? Then $G/H = PGL_2$. How can I get a field $k$ for which $GL_2(k)/Z_{GL_2}(k)$ is smaller than $PGL_2(k)$?