Cisco Anyconnect installation failure

I cannot assist you if you do not provide all the necessary technical details. You have yet to state the precise errors you are encountering. Screenshots of the errors would be useful, also any Cisco AnyConnect logs.

However, most all VPN systems through which you would connect with Cisco AnyConnect will have a configuration profile provided by the VPN Administrator. These are almost always IT managed VPN services from a company or educational institution. It is the VPN Administrators job to upgrade the Cisco AnyConnect version, package the Cisco AnyConnect client and distribute the customized installer that includes the configuration settings along with it.

As an example, if I am to VPN connect to ABC Company, then the company provides the VPN client along with login credentials and perhaps including multi-factor App based authentication. If Cisco AnyConnect fails to install, then it is the VPN IT administrator at ABC Company who needs to fix it. They would have a support contract with Cisco and be able to escalate the problem and receive support from Cisco directly. ABC Company would have an internal help desk or an IT staffer whom you could contact for assistance.

Cisco AnyConnect, Release 4.10 specifically supports macOS Monterey 12.x

https://www.cisco.com/c/en/us/td/docs/security/vpn_client/anyconnect/anyconnect4ten/feature/guide/anyconnect4tenfeatures.html#pgfId-63847

Downgrading AnyConnect will most certainly not work as 4.10 specifically added macOS Monterey 12.x support. Earlier versions will not work with Monterey. There are different 4.10 release builds you may need a newer one.

See this:

https://www.cisco.com/c/en/us/support/docs/smb/routers/cisco-rv-series-small-business-routers/smb5642-install-cisco-anyconnect-secure-mobility-client-on-a-mac-com-rev1.html

What is not working?

Typically, an IT department would be managing Macs with an MDM or other agent driven management tool and they would deploy this App with the company profile included and do it in a fully automated fashion. It's distributed as a DMG file which is an Apple Disk Image and inside the DMG when mounted is a PKG installer and a Profiles folder and perhaps some hidden files in the DMG.

I would normally deconstruct a package like this by mounting the DMG then going to /Volumes/<DMG> in Terminal and listing the files and look for hidden files "ls -la". Next I'll use Suspicious Package to examine the PKG file and look at the scripts and determine what it is doing and how it is doing it. Then I may decide to create a customized PKG with the configurations for the company VPN, etc. Finally stage the custom package on a distribution point and then either push it to the managed Macs or allow the users to pull it from a Self-Service or Customer Portal, etc.

Alternatively, you could copy the DMG to the Mac and have a script do something like this:

hdiutil attach /Users/Shared/anyconnect-macos-4.10.02086-core-vpn-webdeploy-k9.dmg
installer -pkg /Volumes/<DMG>/anyconnect.pkg -target /
hdiutil detach /Volumes/<DMG>

You need to provide more detail. What didn't work? Did you see an error? Are you an IT professional trying to deploy the Cisco VPN software to end users on macOS? Are you an end user trying to install it? Is the VPN for your employer? Did you obtain the latest version of the package from Cisco? Is the package prepared for your VPN by the IT department? There's usually configuration settings that vary between Cisco customers to allow VPN to work. Those configurations need to be provided with the PKG to automatically configure the Anywhere client. Are you being prompted to install a kernel / system extension or allow access to install a network filter. Do you have local admin rights on this Mac?