You might be interested in the English language book, "A History of Muslim Sicily" by Leonard C. Chiaroni.
I warn you it's very long, but it's the best analysis I've ever seen, in English or Italian, for this period of Sicilian and southern Italian history, and was very well received critically. One of the conclusions is that the majority of the people who actually settled in Sicily during that period were Berbers, that it was a male mediated invasion, and that the settlements were on a south/north cline.
As for the DiGaetano et al paper mentioned upthread, rather than being "the" resource about J1e in Sicily, it says almost nothing about it. (Just proof that one should actually read a paper before posting it.) This is the only excerpt I could find about J1 in the paper:
"Haplogroups common both to the European and Eurasian populations are present in Sicily. The most represented are R1b1c-M269 (24.58%), J2-M172 (15.25%) and E3b1a-M78 (11.44%). The co-occurrence of the Berber E3b1b-M81 (2.12%) and of the Mid-Eastern J1-M267 (3.81%) Hgs together with the presence of E3b1a1-V12, E3b1a3-V22, E3b1a4-V65 (5.5%) support the hypothesis of intrusion of North African genes.
7, 12"
The estimates for EM-81, at 2.12%, are actually much lower in this paper than in some others. As to the J1, as I tried to point out upthread, at the time this paper was written they still had not resolved the subclades of J1, so it's impossible to know how much of this actually arrived during the Muslim invasions versus prior periods. I think the same can be said for some of the non EM-81 "E" clades, i.e. I don't know how it can really be said whether they came to Sicily in Neolithic times, during the Metal Ages, or indeed during the Muslim Conquest. We're going to need more resolution of the "E" as well as the "J1" clades, and more sampling and testing for those clades in order to get a better handle on what actually happened. Some old Dna would help, too.