I have an equation of the form $$F_1 * F_2 = a \cdot F_3 - b \cdot F_4$$ where all the $F$'s are cumulative distribution functions and $a$ and $b$ are some constants, and I want to solve for $F_2$. In order to do this, I would like to find the inverse of $F_1$ with respect to convolution, i.e. another CDF $F_1^{-1}$ such that $$F_1 * F_1^{-1} = \delta.$$ However, such inverses do not always exist. Wikipedia has the following unhelpful remark
Some distributions have an inverse element for the convolution
but says nothing about which distributions.
There are a couple of related questions here on MSE here and here, but I don't see any mention of what distributions allow for an inverse for convolution.
So my question is: for what kind of distributions does there exist an inverse element with respect to convolution?