There are some online tools and websites that allow you to view Instagram profile pictures in higher quality. These services usually work by entering the link to the user's profile. However, please note that this may violate Instagram's usage guidelines as you will be leaving the platform.
There are third-party apps that claim to enlarge Instagram profile pictures. You can search for such apps on App Store or Google Play Store and choose the one that suits your needs. Make sure you use trusted apps and check the permissions they request.
Another method is to take a screenshot of the profile picture after enlarging it. To do this, press the power button and the volume button on your smartphone at the same time. This saves the enlarged image to your gallery.
Instagram does not notify the user when you instazoom in on their profile picture.