Private network can not be set up on Ubuntu 20 with Vagrant and VirtualBox

I have a brand new server and I just installed VirtualBox and Vagrant. If I try to set up a private network, I get the following error:

vagrant@zg-2:~/vms/zg2t1$ vagrant up
Bringing machine 'default' up with 'virtualbox' provider...
==> default: Importing base box 'ubuntu/focal64'...
==> default: Matching MAC address for NAT networking...
==> default: Checking if box 'ubuntu/focal64' version '20220311.0.0' is up to date...
==> default: Setting the name of the VM: zg2t1_default_1647172062563_66694
==> default: Clearing any previously set network interfaces...
There was an error while executing `VBoxManage`, a CLI used by Vagrant
for controlling VirtualBox. The command and stderr is shown below.

Command: ["hostonlyif", "create"]

Stderr: 0%...
Progress state: NS_ERROR_FAILURE
VBoxManage: error: Failed to create the host-only adapter
VBoxManage: error: VBoxNetAdpCtl: Error while adding new interface: VBoxNetAdpCtl: ioctl failed for /dev/vboxnetctl: Cannot allocate memory
VBoxManage: error: Details: code NS_ERROR_FAILURE (0x80004005), component HostNetworkInterfaceWrap, interface IHostNetworkInterface
VBoxManage: error: Context: "RTEXITCODE handleCreate(HandlerArg*)" at line 95 of file VBoxManageHostonly.cpp

I'm using the following versions:

Ubuntu 20.04.4 LTS
virtualbox 6.1.32-dfsg-1~ubuntu1.20.04.1
vagrant 2.2.19

My Vagrantfile:

Vagrant.configure("2") do |config|
  config.vm.box = "ubuntu/focal64"
  config.vm.hostname = "zg2t1.softxs.ch"
  config.vm.provider :virtualbox do |v|
    v.customize ["modifyvm", :id, "--name",   "zg2t1"]
    v.customize ["modifyvm", :id, "--memory", "3072"]
    v.customize ["modifyvm", :id, "--cpus", "1"]
    # Basebox ubuntu/xenial64 comes serial port configured, switch it off
    # See https://github.com/joelhandwell/ubuntu_vagrant_boxes/issues/1
    v.customize ["modifyvm", :id, "--uartmode1", "disconnected"]
  end
  config.vm.network :private_network, ip: "172.16.2.2", netmask: "255.240.0.0"
  config.vm.network :forwarded_port, guest: 22, host: 20202
end

Note that if I remove the line config.vm.network :private_network, the VM starts normally. But I need the private network.

I have the following virtualbox modules loaded:

vboxnetadp             28672  0
vboxnetflt             28672  0
vboxdrv               499712  2 vboxnetadp,vboxnetflt

It is suspicious to me, that no vboxnet0 is in the output of ifconfig. What is the problem?