That flute looks to have both a lined socket and a French style tuning slide, where the inner slide either extends partially into the head, or fully. If the latter, you will be able to see metal through the embouchure hole.
To fix the crack properly you’ll have to remove both the socket lining and the inner slide/head lining. Both will require heat, and probably a lot of it, to soften the glue (likely some kind of shellac or hide glue). But the main challenge is going to be how to get a grip on the inner slide when it is hot.
You may need to make some kind of expanding mandrel out of a material that will withstand a great amount of heat. And you’ll probably want to heat it too, otherwise it will cool the slide too much to melt the glue. These kinds of jobs often end up requiring a great degree of brutality, which is best done out of sight of the flutes owner! But they usually work.
Once the liners are removed, clean up and thoroughly dry the wooden part, then glue the cracks, using powdered blackwood and superglue. Then ream both the socket and the receiving hole in the wood for the liner until they fit with now pressure.
When you glue the socket liner and the head liner back in, I would recommend using something like GFlex, which is a flexible epoxy that is also releasable with heat, and hence repairable in the future. But be very careful not to get the new glue on the parts of the tuning slide that you want to move freely.
It can help to coat the parts that don’t want glue with vaseline, and then remove any excess glue before it has fully cured, but after it has ceased to be liquid. That way it doesn’t spread.
Also, fit and clean up the socket liner before you fit and clean up the head liner. Cleaning the excess glue is easier this way. And with the head liner, put the glue in the hole, not on the liner, so the excess squeezes out in a location that allows it to be removed (through the head bore, using a flute cleaning wand with paper towel etc).
Its a tricky job, but it looks like this is a parabolic head (i.e. with a tapered bore) so making a new head would require you to profile the head and make a suitable reamer, which would be very time consuming.
The other strategy, which is much lower cost in terms of effort, but also less of a long term repair, is to simply fill the crack and rely on the existing liners to prevent any leaks. To do this, dry the flute instead of hydrating it, so the crack opens. If then fill the crack with a mix of superglue and blackwood dust. This will have the effect of reducing the pressure on the liners, since you have actually increased the circumference of the wooden head where it passes around the inserted metal parts this way.
If the flute doesn’t leak after doing this, you may be good to go, because the rest of the head may not crack due to the fact that it doesn’t have any metal inside it (here, I’m assuming a partially lined head, which is usually the case).
So, if you do this its simply a matter of filling (not just gluing) the crack and tidying up the exterior. But it has to be done with the flute in its driest state if you want to avoid subsequent opening of the crack.
The crack may have opened on you because you have the flute in a drier environment than it is used to. Keep it humidified.
