Will roof lantern on extension being light to inner room?

[BewareTheBeardedDragon]

My house has an old 70s flat roof extension, and the room behind it has no windows so is reliant on light making its way through from the extension and is quite dark.

The extension room has two large windows, and faces south. It gets a lot of light but not so much gets through to the inner room. There is no way to use the inner room for something that doesn't need so much light, it is the dining area and that cannot be changed.

The flat roof is knackered and needs redoing. I will be beefing up the insulation when this is done and giving it a white coloured roof because the extension room gets very hot in summer. I am wondering whether it might help the inner dark room situation to add a roof lantern on the flat roof when it is redone? But worried that it would be mad to do so given the solar gain problem. I guess that you can get lanterns with solar glass and maybe opening panes?
The extensions windows also need replacing and I will hopefully get solar glass to help with heat issue.

Does anyone know
A) whether a lantern would make any considerable difference to the gloomy inner space
B) if a lantern is a terrible idea on a south facing room that already gets hot?

Thanks

I have a lantern on a south facing extension. It does give light in the inner room but it is also extremely hot in summer. Opening panes make little difference.

If you are going to have one make it as close to the inner room as you can

How much light would you say it gives?
Do you have solar glass on yours @SongforWhoever?

We had a similar dilemma- south-facing flat roofed extension and the inner room has just 2 tiny windows facing next door's wall. Even 2 double patio doors on the extension didn't stop the inner room being dark.
We have just had an exposed pitch put on the roof instead of a flat one, and put 2 veluxes in, one facing East, one West. It has definitely improved the light in the inner room, and in the extension it is far lighter. But a warning, the inner room still isn't a light room, it's just not as dark as it was. We're having thermal blinds on the veluxes for the summer months (not that it ever gets all that hot up here in the Pennines).

( the veluxes open, of course, so in hot weather that will help- I'd not have roof lights in a south facing room/conservatory unless some of them opened)

I think it gives a fair amount of light though difficult to judge as I haven't been in a similar house without the lantern. I don't have solar glass.

We have roof lanterns in our large rear extension and they bring in an amazing amount of light. Our layouts sound similar, ours is open plan but the inner part furthest from the window is the dining area.

We are north facing though so getting too hot is not an issue.

Thank you for all answers.