There are many ways to invest new members and not one definitive one. This is my take on it, so please use the bits you like, make suggestions in the comments and share with your fellow Scouters.
Begin by getting your colony, pack or troop into their horseshoe, as if starting or ending the meeting.
If you use terms like “at attention” then get the pack or troop standing smartly. Beavers can stand still and be quiet for this. Get the lodge, six or patrol leader to bring the new members before you.
Explain that all of the Scouts around the world use the Scout sign as a common symbol and ask all of the invested members to make the sign, together with those who are about to join.
Sometimes I use a flag and sometimes I don’t. It can be useful, held horizontally and NOT touching the floor, to keep the left hand on to stop children wandering if there are quite a few to be invested.
Make the promise with the new members, say a line and then ask them to repeat it. Even if they have learned it, it helps to reduce their stage fright. Remember, there are variations to the core promise in the UK and you should have given the young person and their parent the options beforehand. https://members.scouts.org.uk/supportresources/2454/the-promise?moduleID=10&cat=25,283
Then, I usually ask the new members to turn and salute their fellow beavers/cubs/scouts and for the beavers/cubs/scouts to salute them where we do saluting .
Next, I hand them the badges one by one. We usually have them in a little pack which saves them getting lost so I open one up and start with the World Scout Badge (Purple circle).
I usually use a fact or two from a document like this which I try to make relevant to what’s going on. For example when schools were chatting with astronaut Tim Peake, I would mention that 11 of the 12 people who have walked on the moon were Scouts. A standard line I will use is something like worn by Scouts in every country except five (see link) – places where people like you have much less freedom or hope.
I then hand out the county badge and try to know how many members there are in our county. Then the district badge. Our district badge is next and there is a lot of information about it. Again, I mention that near us, over 1200 people take part in Scouting regularly. Our group nametape is after that, I usually say that we are the largest in the county or that we have been around just a bit longer than their mum and dad. Some of our sections have a nametape and so I mention that is worn by everyone in the hall and then the woggle or patrol badge denotes their lodge/six or patrol. give them a certificate and shake them by their left hand . It seems to work better to do photographs for mum and dad at the end.
My final tip: Try to make investitures memorable for the right reasons and try to make it representative of the fun and adventure we enjoy. Invest them whilst they abseil or in a police cell or on the summit of a mountain.