The first thing to consider is... why you need a group. And how it's different to a page.
A Facebook page is a platform for you to announce things. Imagine a stage, it can be like a Q&A with journalists, but ultimately you are doing most of the talking. It's for broadcasting.
A Facebook group is for discussion. You are in amongst the crowd, and the discussion doesn't have to centre around yourself. In fact, my favourite Facebook groups don't centre around the founder or admins. People will feel free to post topics they are interested in.
When you first start a Facebook group, it could almost be like a page.
Chances are, you will be talking to yourself for a while. The best way to overcome this is by having some early adopters. These can be friends you know in real life, or just people generally interested in the group. People have short attention spans. Woo them to stay, to engage and even to post their own topics. It is really tough when you are just posting post after post with no one else saying anything. After a while it becomes awkward for people to post as they think a) it's just for admin to post b) they don't want to post if no one responds.
So woo these early members.
The easiest way to start a group is to create a splinter group from a bigger group. Bigger groups become unwieldy - they are often very active, but I think they reach a saturation point and then posts get ignored. You will then also have the advantage of having a bigger flow of new members in one go. (It's for this reason we are extremely strict on allowing new members into the Hampstead Mums group as we wish to keep the community as close-knit as possible. In a group of 6.8k members we have 5.6k active members which is astonishing I think for a group that has been going for 4 years). We let in about 20-30% of applicants nowadays, otherwise, we would easily be at 13k members.
So once you've started, to grow the group you need to invite members and have your members invite their friends. I personally never posted about my group in other groups but that's because we were an early adopter. It would be wise to post in bigger groups with a similar demographic as the one you are after. Do bear in mind that different groups have different rules - we don't allow that in Hampstead Mums for example, unless it's a specialist group directly related to them (ie a local charity or SEN group for local parents).
Also, one more point to consider. Public, closed or secret group. I personally would never post in a public group as I don't like to leave too much of a footprint. In reality, it is similar to a Facebook page as you don't need permission to join. Closed groups require permission to join. In terms of security, I would advise members to not think they are in a very "safe place", as it's very limited what admins can do to turf out trolls. And secret groups - these don't show up on searches, and you need to be invited by a member.
And how to grow it.
Really think about what members need, and how you can give it to them.
When I created Hampstead mums, there was no place online for Hampstead parents to discuss whatever parents discuss. The groups would often become platforms for businesses to keep posting adverts. So I banned businesses from posting in the main feed and created a Monday post for businesses to post. This simple alteration allowed the group to grow, and soon other local groups followed suit.
I also introduced a business post in return for a donation to a local food bank. This enabled me to give businesses what they want (exposure on the main feed), raise funds for food banks and not feel like such a plum for doing free work for businesses for nothing.
Anyway, enough of my waffling. Waffle waffle.
Let me know in the comments section if you have any queries about Facebook groups.