We just went through something similar with Evie. Her awake times were always too short and so I worked on extending them to 1.5 hours at the same time I started extending her naps. What really helped for me was to see what time she woke up then write down the time of her first feed. Then I would write down the rest of the times for feeds based on a 3 hour schedule which is what we were trying to acheive (you would do this for 3.5 or 4 hours probably since most 5 month olds can handle 2 hours of awake time but Evie couldn't at first). So if she woke up and ate at 7am, I would write down, feed times of 7, 10, 1, 4, and 7pm. Then I had an easier time keeping track and keeping her on schedule. So to start, I was trying to keep her awake for 1.5 hours. She'd wake up and eat at 7am, then I'd know she must be asleep by 8:30 or else she would be too fussy (if late) or not tired enough (if early). If your little one takes longer to settle, you have to keep that in mind and work it into the activity time so you are not putting them in the crib too late. Once asleep, I watched for the 45 min mark and was there to pat/shush her back to sleep, knowing that I had until the next feed to get some good sleep in. So I'd be saying to myself, she's awake and it's only 9:15 but she doesn't eat until 10 so we have time to get some more sleep in (rather than getting her up and dealing with tired baby or feeding too soon). I tried wake to sleep to extend the naps but it didn't work for us. That's why I did the pat/shush. She would often wake up all smiles but I knew she hadn't had enough sleep if it was only 20-45 mins. So I would put one hand on her tummy and the other sort of under her legs (I found by trial and error that this position is what worked for us since she couldn't push her bum up off the crib like a crab and scoot around which just kept her from settling) and I would close my eyes and say "shhhh, shhhh, shhhh". She always fussed and fought and cried for a bit but then she would settle and go back to sleep. I think it took about half an hour the first time and then gradually got less and less until now she often doesn't even need me in there, she just puts herself back to sleep. Of course, any sound or interruption makes you have to start all over, but as long as you are consistant, it pays off. The reason I had to finally do this was because her short naps were interferring with her nighttime sleep all of a sudden. I'm happy to say the last three days she has had 1hour to 1.5 hour naps with very little intervention on my part and has slept through the night all three nights!
Hope that helps (keep in mind you would be working toward a 3.5 or 4 hour schedule though- we'll be doing that next.)