The local authority does not take on the lead responsibility for welcoming and supporting the refugee family. That is your job. But you need the council on your side.
It is obvious why, when you think about. For a start, they are the gateway to accessing many services in your area – in particular school places for the refugee children. Secondly, if anything went wrong with your sponsorship (thankfully, that rarely happens) the local authority would have to assume responsibility for the welfare of the refugee family. All this means that you should develop a good relationship with your local authority and will have to obtain their written consent for your group to sponsor a family.
More details in the WINNING THE CONSENT OF YOUR LOCAL AUTHORITY section on the Act page.