This petition was rejected

The Government e-Petitions Team gave the following reason:

There are no proposals to integrate the UK armed forces with European armed forces.

In a recent answer to a parliamentary question, the Government stated:

"The Withdrawal Agreement and Political Declaration provide the option, but no obligation, for the UK to continue to contribute to CSDP operations and missions on a voluntary basis, and where of benefit to UK interests. This does not undermine the UK's sovereignty, our command or control of our Armed Forces nor does it oblige us to participate in a 'defence union' or any EU defence initiatives.

"The UK will retain full sovereign control over its defence, intelligence services and decision-making after leaving the EU. The UK will also retain control over the deployment of its Armed Forces and their equipment. Any future security partnership negotiated with the EU would reflect this position."

You can read the full answer here: questions-statements.parliament.uk/written-questions/detail/2020-01-15/HL456

You can find out more about the Permanent Structured Cooperation here: commonslibrary.parliament.uk/research-briefings/cbp-8149/