First off, you are never going to get an objectively "right" answer to this one. But let me share my opinion based on my own experience as a person with Asperger's (now known as autism-spectrum disorder).
Back in 2012 a good friend of mine presented me with the Wikipedia page for Asperger's syndrome and told me she thought it reminded her of me. This was done privately at work, we were in neighbouring cubicles, and she then left me to my own to read the article at my own pace. It put me down a bit but I thought about it a bit for the next couple of months. Roughly six months later, for completely other reasons, I met with a psychiatrist to evaluate a possible burn-out.
I got my preliminary diagnosis about a month after that, and the on-paper diagnosis about three months after that.
I have now spent about four years and a bit having to redefine my entire world-view to one where I and the average person have a very different experience of the world. Especially in interpersonal relations. I would liken it to a religious person losing their faith. Scary stuff.
Should you let your friend know? I would say absolutely yes. All I can say is that I am deeply grateful to my friend for caring and trusting me enough to let me know her thoughts, and the diagnosis is the best thing that has ever happened to me.
Yes, it is scary. Yes, it is hard. Yes, it can potentially cause problems. Yes, you can end up in a lot of grief depending where you live. But frankly, the realisation and wealth of research material and personal anecdotes from other autistic people have helped to answer a whole heap of questions I have had for years.
And I now have access to a whole heap of aid that has taken a load of my shoulders and I can avoid large social gatherings at holidays without hurting anyone's feelings (I don't do well in those and mainly sit in a corner in a panic state the entire event).
Remember to adjust the way you suggest it based on your friends personality. I am theoretically-minded and the first books I picked up to were course material for psychiatrists and poured over stuff from Tony Attwood. Another autistic I know do not respond well to theoretical books and struggled with all theoretical courses in school, but excelled in more practical courses, for such a person presenting a Wikipedia page or theoretical book will likely not cause a good reaction.
For any personality, and anyone in general, I highly recommend the excellent read The Reason I Jump by Naoki Higashida (ISBN: 978-0-812-99486-5).
Another thing to keep in mind when discussing this with your friend is that, sadly, the cultural views relevant to you will have an affect as well. I live in Sweden where, while people might not always be comfortable discussing mental health, there is generally a good understanding and social support for special needs with several law frameworks to rely upon. I understand that it is the same in Germany, but I do not know about Switzerland. So keep that in mind as well.