How to add permissions on custom column

Hard to answer that question without knowing the version used, what type of permissions you have added, what operation is failing exactly and where, which authenticator you have used, how you have created the custom permission group etc.

We do have a complete video series on our YouTube channel dealing with the different permission options. It might be worth a look https://www.youtube.com/watch?v=duPowKKmusA&list=PL242czeZwlAl46Xv0UnBDwTm-8VN5p4qI 

Answers to your questions

1. We are using version 8.0

2. We created a custom permission group of VI_4_AllManager and assigned this permission group to a custom column. For this custom column, we had checked "Can View" and "Can Insert" which means, user should be able to insert the data in custom column if there is no data available but if data is there, it should not be editable. This same permission is working for OOTB column but not for custom column.

3. Created custom permission group using the wizard available

Just for clarification, CanInsert for a column is saying you can modify this column during an INSERT of the object.

And, did you assign your new permission group to an application role?

yes so I want this same functionality when user inserts the object, it should be allowed to insert data in that specific custom column. But once added and saved, it should not be allowed to edit this custom column and yes I assigned this permission group to a application role

Did you set an InsertWhereClause for the table in question? (1=1 is okay as well).

Did you test this in ObjectBrowser with the correct authenticator and checked the permission group assignments for the logged in user?