Brain Twisting

Let the number be $\overline{ABCDE}.$

First, we know that $B=2A,$ and $E = A+B = A + 2A = 3A.$

Since $E$ is a digit divisible by 3, it has to be either 3, 6, or 9. But it cannot be 3, since $A$ cannot be 1 since none of the digits are 1. It also cannot be 9, since $C$ is the largest digit and no digits are larger than 9.

Thus, $E$ must be 6, giving $A=2$ and $B=4.$

Now, we know that $D = \frac{1}{4} (A+B+C+D+E) = \frac{1}{4}(2 + 4 + C + D + 6) = 3 + \frac{1}{4}C + \frac{1}{4}D,$ so $C = 3D -12.$ Since $3D+12$ is divisible by 3, $C$ must be divisible by 3. But since $C$ is the largest of the digits and $E=6,$ we must have $C=9.$ Since $C=9,$ $D = \frac{9 +12}{3} = 7.$

Thus, the 5-digit number is $24976.$

$\textbf{First of all}$ .. you really need to state your counting direction, meaning the First number is it the one on the far Left or the far right?!, otherwise every one will use his own default for me the first always represents the units

$\textbf{Second:}$ the solution

$\text{Assign letters from a to e for the unknown digits as follows}$

$\Large\color{maroon}{\textbf{edcba}}$

$e>1 \dots\dots \color{#3D99F6}{eq(1)}$ $d=2e \dots\dots \color{#3D99F6}{eq(2)}$ $c>a>d \dots \color{#3D99F6}{eq(3)}$ $c>b \dots\dots \color{#3D99F6}{eq(4)}$ $a=d+e=3e \dots \color{#3D99F6}{eq(5)}$ $\large b=\frac{(a+b+c+d+e)}{4}=\frac{(c+b+6e)}{4}$

$4b-b=c+6e\space\dots\space 3b=c+6e$
$\large b=\frac{(c+6e)}{3} \dots \color{#3D99F6}{eq(6)}$

using eq(3) & eq(5) $\color{#3D99F6}{c>3e}$ $\textbf{If e=2}$ $\text{c>6 ,,, "c" could be 7,8,9}$ $\textbf{If e=3}$ $\text{c>9 ,,, error 9 is the largest digit}$

$\textbf{so}$ $e=2,, d=4,, a=6$

Using equation (6) $\textbf{so}$ $c=9$ $\textbf{and}$ $b=\frac{(9+12)}{3}=7$

$\textbf{Finally the number is=}\large\boxed{\color{maroon}{24876}}$