When are all the convolution roots of an infinitely divisible probability measure infinitely divisible? Let $G$ be a topological group. 
Let us say that a probability measure $\mu$ on $G$ is strongly infinitely divisible (SID) if $\mu$ is infinitely divisible and any probability measure $\nu$ on $G$ such that $\nu^{*n}=\mu$ for some natural $n$ is also infinitely divisible. 
Let us say that a probability measure $\mu$ on $G$ is super-strongly infinitely divisible (SSID) if $\mu$ is infinitely divisible and for any probability measure $\nu$ on $G$ such that $\nu^{*n}=\mu$ for some natural $n$ there is a one-parameter convolution semigroup $(\nu_t)_{t\ge0}$ of probability measures on $G$ such that $\nu_1=\nu$. 
Clearly, every SSID probability measure on $G$ is SID. Is it true that, vice versa, every SID probability measure on $G$ is SSID? 
If this statement is true, will it remain true if "one-parameter" is replaced by "continuous one-parameter"? 
This question is related to, and motivated by, the question here. 
 A: I suppose that your notions of "strongly" and "super-strongly" infinitely divisible are closely related to the class $I_0$ of probability distributions whose all components (in the sense of convolution) are infinitely divisible.
The fundamental theorem of Khinchin on decompositions says that every probability distribution
is a convolution of (possibly infinitely many) indecomposable distributions and a distribution of the class $I_0$.
(A distribution is called indecomposable if it is not a convolution of two non-trivial
distributions).
This class $I_0$ was studied a lot, at least for the case when the group $G$ is the real line or $R^n$. Most of the results are contained in the book 
Yu. Linnik and I. Ostrovski, Decomposition of random variables and vectors,
English translation: AMS 1977.
I do not think that a complete parametric description of this class $I_0$
is available, but it is available under some mild additional conditions.
Many of these results were generalized to Abelian groups in
Felʹdman, G. M., Arithmetic of probability distributions, and characterization problems on abelian groups. AMS, 1993.
