No group can have a common difference $\ge 4$ as there is no number $\ge 13$
If the group containing $1$ has the common difference $=1$ i.e. if one group is $\left(\mathrm{1,2},\mathrm{3,4}\right)$, there will be two ways to group others - $\left(\mathrm{5,6},\mathrm{7,8}\right)$, $\left(\mathrm{9,10,11,12}\right)$ and $\left(\mathrm{5,7},\mathrm{9,11}\right),\left(\mathrm{6,8},\mathrm{10,12}\right)$.

If the group containing $1$ has the common difference $=2$ i.e. one group is $\left(\mathrm{1,3},\mathrm{5,7}\right)$, there will be one way to group others $\left(\mathrm{2,4},\mathrm{6,8}\right),\left(\mathrm{9,10,11,12}\right)$.

If the group containing $1$ has common difference $=3$ i.e. if one group is $\left(\mathrm{1,4},\mathrm{7,10}\right)$ there will be one way to group others $\left(2,5,8,11\right),\left(3,6,9,12\right)$

$\Rightarrow$ The number of ways $=2+1+1=4$