The list of the different religious groups can be found here in Swedish, but to mention some:
The Church of Sweden
The Roman Catholic Church
The different Orthodox churches
The free churches, including e.g. Pentecostals, Seventh-Day Adventist, Methodists, the Salvation Army and some Lutheran organisations
The Jehovah's Witnesses
DUNNO:There is also the question what to do with consciencious objectors, i.e. pastors who refuse to marry same-sex couples.
The Scientology Church
Some Lutheran organisations
In many of the above religious groups, there is an ongoing discussion about the right to perform marriages - should our church retain that right or give it up? Does the right to perform marriages include an obligation for the individual pastor to do so? Should the right to perform marriages no longer be a question for a whole denomination, but rather given individually to those pastors who have no objections to marrying anyone that the state deems worthy? These questions are far from being solved.