Change email blog posts to summary

I’m using Jetpack as a subscribe option for my blog. I also have a custom theme. I have noticed that my email subscribers get the generic email of the full blog post even though I have “summary” selected in settings > reading. I would like for my readers to have to click on the link to read more since there are no share icons on the generic email post they receive. Wondering why it is showing the whole post in the email? Also, is there a plugin that will allow for more customization of the subscriber emails or make them look nicer?