"Individuals with ASD [autism spectrum disorder] and their relatives are at increased risk of ADHD [attention-deficit hyperactivity disorder]."
So said the paper published by Laura Ghirardi and colleagues  (open-access) who studied "1 899 654 individuals born in Sweden between 1987 and 2006" and identified some 28,000 cases of ASD and 82,000 cases of ADHD "with 13 793 individuals... being comorbid cases."
Results: "Individuals with ASD were at higher risk of having ADHD, compared with those who did not have ASD (OR=22.33, 95% CI: 21.77–22.92)." This is not a particularly surprising finding given what is already emerging in peer-reviewed science and clinical circles about autism rarely appearing in some sort of diagnostic vacuum (see here). The authors also report that: "the prevalence of ASD–ADHD comorbidity was higher in males (1.01%) than in females (0.43%)."
Looking further at various degrees of familial relationship starting with monozygotic (MZ) twins - twins sharing the same genome - outwards to full- and half-cousins, authors reported a sort of sliding scale of risk on the association between autism and ADHD. So: "monozygotic twins of ASD cases had an increased risk of having ADHD" which was a higher risk than that noted for dizygotic (non-identical twins). Next on the list of heightened risk of ADHD where another family member was diagnosed with autism was a full sibling (actually, the calculated risk was slightly higher for full siblings over dizygotic twins) and then maternal and paternal half siblings respectively. The authors note that: "The pattern of within-family associations between ASD and ADHD across different types of relatives reflected the decreasing genetic resemblance among them." This potentially signifies quite a substantial role for genetics when it comes to the co-occurrence of autism and ADHD as per other important data  (assuming that environmental - non-genetics - factors don't impact on pregnancy or before pregnancy processes related to sperm and egg for example).
Where next? Well, the pathway(s) potentially linking autism and ADHD still requite a lot more investigation. With my continued interest in how autism and ADHD might 'set someone up' for later mental health issues (see here and see here for examples) I'd be inclined to widen the range of diagnoses/labels that might also need examination pertinent to processes involved. Assuming also that genes for autism or ADHD might not necessarily be just genes for autism and ADHD (see here), an open mind needs to be kept of what could be potentially involved, as per the whole immune system link for example (see here). I'd also be inclined to suggest that the Ghirardi findings suggest greater examination a possible new dimension when it comes to concepts like the broader autism phenotype (BAP) (see here) too (including signs and symptoms based around sensory profiles ).
Music to close: The Flaming Lips - She Don't Use Jelly.
 Ghirardi L. et al. The familial co-aggregation of ASD and ADHD: a register-based cohort study. Mol Psychiatry. 2017 Feb 28.
 Sokolova E. et al. A Causal and Mediation Analysis of the Comorbidity Between Attention Deficit Hyperactivity Disorder (ADHD) and Autism Spectrum Disorder (ASD). J Autism Dev Disord. 2017 Mar 2.
 Bijlenga D. et al. Atypical sensory profiles as core features of adult ADHD, irrespective of autistic symptoms. European Psychiatry. 2017. Feb 21.
Ghirardi L, Brikell I, Kuja-Halkola R, Freitag CM, Franke B, Asherson P, Lichtenstein P, & Larsson H (2017). The familial co-aggregation of ASD and ADHD: a register-based cohort study. Molecular psychiatry PMID: 28242872