You want to split your input on space, not tab.

e.g.

       # my @cols = split(/\t/); # Change this
       my @cols = split(' ');  # To this.

BTW, you want:

tr " " "\t" < namconvmars.txt

You do not need three "@group"s -- you either need three scalars ($field0, $field1, $field2) or one @group, which you could print with join("\t",@group);

A simpler solution is to:

while (my $line = <$input>) { 
  chomp($line)
  print join("\t",split(/\s+/,$line),"\n"
}

or

$line =~ s/\s+/\t/;
print $line

There are some Perl-ings you need to understand first:

my $infile = $ARGV[0];

This line reads the first command line argument after your script name and pass to the variable $infile

open (my $infile, "<", "namconvmars.txt")
or die "Can't read from $infile: $!";

You are declaring again $infile (that is what my does), also you are reusing the variable to be a file pointer. So, you don't need the first line my $infile = $ARGV[0] because you never used it.

my (@group1, @group2, @group3);

while (<$infile>){
    my @cols = split(/\t/);
    push @group1, @cols[0];
    push @group2, @cols[1];
    push @group3, @cols[2];
    print "@group1\t@group2\t@group3";
}
close $infile

On this part I think you want to collect the values, but if your intention is to simply convert each line, you don't need the arrays, just read, modify and print each line. The complex part I see, when you split the line using spaces, the second element is splitted too ("Ana Biba" -> ["Ana", "Biba"], to avoid this you will need to reconstruct that element. Something like:

#!/usr/bin/perl
use strict;
use warnings;
my $file = "namconvmars.txt";
open (my $infile, "<", $file)
or die "Can't read from $file";

while (<$infile>){
    my @cols = split(/\s+/, $_);  # break line using spaces
    my $first = shift(@cols);  # grab first element
    my $last  = pop(@cols); # grab last element
    my $mid   = join " ", @cols; # reconstruct middle element
    print join "\t", $first, $mid, $last;
}
close $infile

awk 'BEGIN{FS=" ";OFS="\t"}{print $1,$2" "$3,$4}' in > out

edit. More general solution where the first and last space are replaced with tabs

awk 'BEGIN{FS=" "}{L=$NF; NF--; sub(" ","\t",$0); print $0"\t"L}' in > out

with sed:

$ sed 's/\s\+/\t/g' test.txt          
AB  Ana Biba    1029293.34341