It would depend on your relationship with that person. If you are very close friends, then it would probably be permissible to call her Liz or Lizy. It also depends on what they prefer to be called. I usually don't use nicknames though, unless someone requests that I call them by a different name.

The general rule with nicknames is to ask people what they want to be called, and then call them that. If you don't know, start with the full name ("Elizabeth") and then let her say "Call me ____." Some women named Elizabeth *hate* Lizzy. Some people love it. Also, if someone tells you not to call them by a nickname, it would be very rude to continue to do so. Other nicknames for Elizabeth are: Eliza, Liz, Liza, Lizbeth, Betty, Beth, Bet, Ellie, Libby, Bess, Betsy, (and many others, it is a very common name, it has a lot of nicknames). But, again, any grown woman knows which (if any) of those names she likes - she'll tell you which one (if any) to use.

The important thing with nicknames in any culture is the relationship. It's similar to the formalities used in Japanese. With close friends you might drop the honorific さん for example. The main difference though is that in many English speaking cultures it would be rare then to refer to the person by their full name except when you're making fun of them or if you're in a very formal setting such as a business meeting. But even at weddings or funerals the person would still be referred to by their nickname. As far as Lizzy as a nickname is concerned it varies from person to person. Lizzy is a youthful nickname and some people like to hold on to that nickname for as long as possible. Some prefer to have a less fun or carefree sounding nickname and would use "Liz" to sound more mature. It's rare to find a young lady called Liz though as it might make them feel old. Other possible nicknames for Elizabeth are "Elle" or "Beth"

I found out that should not decide their nickname on my own. In Japan, we often decide each other's nickname. It is interesting and thank you for your help:)

It wouldn't necessarily be rude, but it might be awkward. It kind of depends on the person. You could call her "Liz." That is a more popular nickname. Usually, if someone has a nickname thst they like, they'll just introduce themselves with that name. Most people don't really give people nicknames much unless they are your family members, close friends, or partners. Hope this helps ~~