  We answer an open problem raised by Chen and Zhang in 2008 and prove that,
for any minimal projective 3-fold $X$ of general type with the geometric genus
$\geq 5$, $X$ is birationally fibred by a pencil of $(1,2)$-surfaces (i.e.
$c_1^2=1$, $p_g=2$) if and only if the $4$-canonical map $\varphi_{4,X}$ is
non-birational. The statement does not hold for those with the geometric genus
$\leq 4$ according to our examples.
