As I mentioned in a comment above, you want to classify monic polynomials of degree two over a field $K$ of characteristic two under the automorphism group of $K[X]$, which is that of maps $X\mapsto \alpha X+\beta$ with $\alpha$, $\beta\in K$ and $\alpha\neq0$.

If the polynomial is reducible, then it is easy to see that it is in the orbit of $X^2$ or of $X(X+1)$, and these two are not in the same orbit.

Suppose then that the polynomial is $X^2+aX+b$ and that it is irreducible.

If $a=0$, then the polynomial is $X^2+b$, with $b$ not a square, and two such polynomials with $b$, $b'$ nonsquares are in the same orbit iff they differ in a square.

If $a\neq0$, the change of variable $X\leadsto aX$ shows that the polynomial is in the orbit of one of the form $X^2+X+b$. Let $\wp:t\in K\mapsto t^2+t\in K$. The polynomial $X^2+X+b$ is irreducible iff $b$ is not in the image of $\wp$, and it is in the same orbit as $X^2+X+b'$ iff $b$ and $b'$ differ by an element in the image of $\wp$.

We conclude that the orbits are: (i) two special ones, (ii) one for each nonsquare of $K$ up to addition of squares and (ii) one for each nonimage of $\wp$ up to addition of elements in the image of $\wp$.

This is as far as you are going to get without being more specific about the field (because the arithmetic of the field enters as soon as you want to describe what is a square and what is not, and what is in the image of $\wp$ and what is not) This has nothing to do with the characteristic, and you would have exactly the same problem over any field: classifying irreducible polynomials of degree two is an arithmetic problem.

As I mentioned in a comment above, you want to classify monic polynomials of degree two over a field $K$ of characteristic two under the automorphism group of $K[X]$, which is that of maps $X\mapsto \alpha X+\beta$ with $\alpha$, $\beta\in K$ and $\alpha\neq0$.

If the polynomial is reducible, then it is easy to see that it is in the orbit of $X^2$ or of $X(X+1)$, and these two are not in the same orbit.

Suppose then that the polynomial is $X^2+aX+b$ and that it is irreducible.

If $a=0$, then the polynomial is $X^2+b$, with $b$ not a square, and two such polynomials with $b$, $b'$ nonsquares are in the same orbit iff they differ in a square.

If $a\neq0$, the change of variable $X\leadsto aX$ shows that the polynomial is in the orbit of one of the form $X^2+X+b$. Let $\wp:t\in K\mapsto t^2+t\in K$. The polynomial $X^2+X+b$ is irreducible iff $b$ is not in the image of $\wp$, and it is in the same orbit as $X^2+X+b'$ iff $b$ and $b'$ differ by an element in the image of $\wp$.

We conclude that the orbits are: (i) two special ones, (ii) one for each nonsquare of $K$ up to addition of squares and (ii) one for each nonimage of $\wp$ up to addition of elements in the image of $\wp$.

This is as far as you are going to get without being more specific about the field (because the arithmetic of the field enters as soon as you want to describe what is a square and what is not, and what is in the image of $\wp$ and what is not) This has nothing to do with the characteristic, and you would have exactly the same problem over any field: classifying irreducible polynomials of degree two is an arithmetic problem —the only difference is that in odd characteristic you need only care about the function $x\mapsto x^2$ while in even characteristic you have to worry also about the map $\wp$. You'd have a similar outcome if you try to classify cubic polynomials in characteristic three.

As I mentioned in a comment above, you want to classify monic polynomials of degree two over a field $K$ of characteristic two under the automorphism group of $K[X]$, which is that of maps $X\mapsto \alpha X+\beta$ with $\alpha$, $\beta\in K$ and $\alpha\neq0$.

If the polynomial is reducible, then it is easy to see that it is in the orbit of $X^2$ or of $X(X+1)$, and these two are not in the same orbit.

Suppose then that the polynomial is $X^2+aX+b$ and that it is irreducible.

If $a=0$, then the polynomial is $X^2+b$, with $b$ not a square, and two such polynomials with $b$, $b'$ nonsquares are in the same orbit iff they differ in a square.

If $a\neq0$, the change of variable $X\leadsto aX$ shows that the polynomial is in the orbit of one of the form $X^2+X+b$. Let $\wp:t\in K\mapsto t^2+t\in K$. The polynomial $X^2+X+b$ is irreducible iff $b$ is not in the image of $\wp$, and it is in the same orbit as $X^2+X+b'$ iff $b$ and $b'$ differ by an element in the image of $\wp$.

We conclude that the orbits are: (i) two special ones, (ii) one for each nonsquare of $K$ up to addition of squares and (ii) one for each nonimage of $\wp$ up to addition of elements in the image of $\wp$.

This is as far as you are going to get without being more specific about the field (because the arithmetic of the field enters as soon as you want to describe what is a square and what is not, and what is in the image of $\wp$ and what is not)

As I mentioned in a comment above, you want to classify monic polynomials of degree two over a field $K$ of characteristic two under the automorphism group of $K[X]$, which is that of maps $X\mapsto \alpha X+\beta$ with $\alpha$, $\beta\in K$ and $\alpha\neq0$.

If the polynomial is reducible, then it is easy to see that it is in the orbit of $X^2$ or of $X(X+1)$, and these two are not in the same orbit.

Suppose then that the polynomial is $X^2+aX+b$ and that it is irreducible.

If $a=0$, then the polynomial is $X^2+b$, with $b$ not a square, and two such polynomials with $b$, $b'$ nonsquares are in the same orbit iff they differ in a square.

If $a\neq0$, the change of variable $X\leadsto aX$ shows that the polynomial is in the orbit of one of the form $X^2+X+b$. Let $\wp:t\in K\mapsto t^2+t\in K$. The polynomial $X^2+X+b$ is irreducible iff $b$ is not in the image of $\wp$, and it is in the same orbit as $X^2+X+b'$ iff $b$ and $b'$ differ by an element in the image of $\wp$.

We conclude that the orbits are: (i) two special ones, (ii) one for each nonsquare of $K$ up to addition of squares and (ii) one for each nonimage of $\wp$ up to addition of elements in the image of $\wp$.

This is as far as you are going to get without being more specific about the field (because the arithmetic of the field enters as soon as you want to describe what is a square and what is not, and what is in the image of $\wp$ and what is not) This has nothing to do with the characteristic, and you would have exactly the same problem over any field: classifying irreducible polynomials of degree two is an arithmetic problem.