By using agetty

How

Install agetty:

# apk add agetty

Edit /etc/inittab to use agetty
Example for the virtual terminal tty1:
tty1::respawn:/sbin/agetty --autologin root tty1 linux

Example inittab entry for a serial terminal on ttys01:
ttyS0::respawn:/sbin/agetty --autologin root ttyS0 vt100

Tip: You can change the `tty1` or `ttyS0` to a different serial port or virtual terminal as you please. `root` can be changed to a different user as well. Finally the terminal type (`linux` and `vt100` in our examples) can be changed to a wide variety of serial terminals.

By compiling your own autologin wrapper

How

Writing a wrapper, called autologin, around /bin/login and moving it in /usr/sbin/
Editing /etc/inittab specifying the use of /usr/sbin/autologin instead of /bin/login

Prerequisites

A C compiler (e.g. gcc or tcc)
The musl-dev package which contains the C standard library

Example on how to assolve the prerequisites:

# apk add gcc musl-dev

Writing the autologin.c program

Create a file in this example called autologin.c

Contents of autologin.c

#include <unistd.h>

int main() 
{
    execlp( "login", "login", "-f", "root", 0);
}

The program makes a system call to execute the login binary (part of busybox) which will be searched in $PATH.

As parameters are passed:

-f flag which stands for "Do not authenticate (user already authenticated)"
username in this example is root but if you created a new user, its username can be used instead.

Compiling the autologin.c program

If using gcc:

# gcc -o autologin autologin.c

Move the binary autologin to /usr/sbin

# mv autologin /usr/sbin/

Editing /etc/inittab

Open /etc/inittab

replace ":respawn:/sbin/getty" with ":respawn:/sbin/getty -n -l /usr/sbin/autologin" for each TTY you want to enable autologin.

The getty's -n flag do not prompt the user for a login name
The getty's -l flag invokes a custom login instead of /bin/login; in our case it is set to invoke /usr/sbin/autologin

Note

To perform such a replacement on all TTYs, the following command can be used:

# sed -i 's@:respawn:/sbin/getty@:respawn:/sbin/getty -n -l /usr/sbin/autologin@g' /etc/inittab

"@" is used as a delimiter
The -i flag edits the file in-place

Cleaning up

It is possible to remove the autologin.c file, the C compiler and the musl-dev package

TTY Autologin