The government coalition and cabinet has little to do with the makeup of the Knesset and the laws passed by the MKs. The MKs are part of parliament and are lawmakers. Each MK tries to pass laws in ways that he or she thinks will improve Israeli society, generally according to the worldview of each specific MK.
One of the more contentious ideas for a law with religious ramifications has been the inclusion of women on the committee that appoints judges to the religious courts, the beis din. The haredi parties have been very upset about such suggestions, while the more liberal parties have demanded it. In recent years, since the issue became big, the advancement of such an idea has always been subject to political wrangling. As long as the Haredi parties have been strong with a lot of political influence, they have been able to suppress such a law from passing granting women the power to appoint dayanim.
Now the haredi parties seem to have less political clout, and the more liberal parties have been on the upswing. We are most likely on track to see a lot of more liberal religious laws being passed, or proposed at least.
One such law proposal is regarding the inclusion of women on the committee for appointing judges to the religious courts. yesterday, MKs from HaBayit HaYehudi and Yesh Atid teamed up to propose a law. of course there is no guarantee that they will succeed in passing this, or any other similar, law just because their political power is on the upswing, but their chances are better now than at any other time in recent years.
The law proposed by MKs Shuli Muallem Rafaeli (BY) and Aliza Lavi (YA) is to include four women on the committee.
The MKs explained their law proposal by saying that the rabbinical courts are responsible for the standing of Jews in the Jewish country. The courts are made up of male judges, as per the halacha and has traditionally been so for ages. The courts though deal with both men and women equally. As a modern society, open and democratic, there is the right for equality, but the make-up of the courts do not reflect that due to their very nature. The only way to allow women to influence the style of the rabbinical courts is via the committee for appointing the judges - and while there have been women in the past on the committee, currently no women serve on it..
So, they have proposed that 50% of the members of the committee would be women, and in addition there would be an 11th member appointed to the committee who would be a to'enet rabbanit.
(source: NRG)
There is no guarantee that the law will pass, just because right now it looks they have increased their power. There will be some other MKs who will support it because they agree with it. There will be some who will oppose it, because they disagree with it. The politics come into play by persuading the MKs who don't have a predisposition one way or the other to the idea - convincing them which way to vote is often dependent on politics and coalitions. If BY and YA get into the government coalition, they can use voting support on such bills as a factor in their favor. If they do not, bills such as these will have little chance of passing.