There are probebly people on her that have different ideas for you, but I would start on the weekend and just stay outside with her untill she goes and then treat and praise her big time. Keep doing this both days, then on the week days crate her like you have and try and get her to go potty.
Now this is where someone might have a better idea. If I were you I would crate her again even if she did not go outside on your break from work. She will go a few times in the crate but will learn within a week or two that the time to go is outside. At least she is 4 months and has some control. Dogs don't like to go where they are going to lay or be so she will learn to not go in the crate and go outside. It will not be fun cleaning the crate, but with the same routine, praise, and treats she will eventually learn. Also feed her all meals in her crate for her to learn it is a good place and that dogs also don't like to go potty where they eat.
*We are in the process of potty training our puppy.I don't want to jinx us but just the last three days she has not gone in her crate! We used the method above. She went in the crate and it wasn't fun cleaning her and the crate everyday but it makes you so proud and happy when she learns.