That statistic - turns out is never tracked - but surely does feed Islamaphobia hunger.
"Wherever they get that number from, it's not from this report since it generally doesn't break down the population in other groups than those born in Sweden with parents born in Sweden, those born in Sweden with one or both parents born outside Sweden (these two groups are sometimes tracked separately), and those born outside Sweden. Religion is not tracked at all.
I should also note that this report is on crime in general and not specifically on rape.
Here's a relevant quote. The translation is mine and quite hastily done and with limited knowledge of the legal terms in English, so take it for what it's worth.
"En typ av brott som är föremål för stor uppmärksamhet i den allmänna debatten i dag är sexualbrotten. Det finns skäl att här lyfta fram att andelen av de utrikes födda, som är misstänkta för ett sexualbrott under perioden, är mindre än en halv procent och andelen som misstänks för våldtäkt och försök till våldtäkt är mindre än 0,3 procent."
"One type of crime that has received a lot of attention in public discussions are sexual crimes. It is relevant to point out that the portion of people born outside Sweden who have been suspected of a sexual crime during the period is less than half a percent, and the portion suspected of rape or attempted rape is less than 0.3 percent."
One comparison from the table on p. 66 is that among men born 1945-1981 who are born in Sweden with both parents born in Sweden, 2.1 per 1000 were suspected of rape one or more times during 1997-2001, while the same number for those born abroad was 5.7. However, since there are many more times as many people in the first category, and the second category also contains many Europeans who are unlikely to be Muslims, the number 77.6% is clearly false even when looking at the data.
As for where this piece if (dis-)information comes from, the most likely source is one of the anti-immigration groups who want to portray immigrants and refugees in general and Muslims in particular as criminal monsters who shouldn't be let in.
The fact is that according to this report over 95% of those born abroad have not been suspected of any kind of theft or "crime against person", which includes all violent crimes. (If you include all crimes the number is about 87%. My guess is that a large portion of the increase is explained by suspicion of crimes against immigration laws but that's just my guess.)\
https://www.quora.com/Sweden-Is-it-true-that-77-6-of-rapist-in-Sweden-are-Muslim